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Mathematics LibreTexts

9.E: Sequences and Series (Exercises)

9.1: Sequences

Q9.1.1

Find the first six terms of each of the following sequences, starting with \(\displaystyle n=1\).

1) \(\displaystyle a_n=1+(−1)^n\) for \(\displaystyle n≥1\)

Solution: \(\displaystyle a_n=0\) if \(\displaystyle n\) is odd and \(\displaystyle a_n=2\) if \(\displaystyle n\) is even

2) \(\displaystyle a_n=n^2−1\) for \(\displaystyle n≥1\)

3) \(\displaystyle a_1=1\) and \(\displaystyle a_n=a_{n−1}+n\) for \(\displaystyle n≥2\)

Solution: \(\displaystyle {a_n}={1,3,6,10,15,21,…}\)

4) \(\displaystyle a_1=1, a_2=1\) and \(\displaystyle a_n+2=a_n+a_{n+1}\) for \(\displaystyle n≥1\)

Q9.1.2

5) Find an explicit formula for \(\displaystyle a_n\) where \(\displaystyle a_1=1\) and \(\displaystyle a_n=a_{n−1}+n\) for \(\displaystyle n≥2\).

Solution: \(\displaystyle a_n=\frac{n(n+1)}{2}\)

Q9.1.3

6) Find a formula \(\displaystyle a_n\) for the \(\displaystyle nth\) term of the arithmetic sequence whose first term is \(\displaystyle a_1=1\) such that \(\displaystyle a_{n−1}−a_n=17\) for \(\displaystyle n≥1\).

Q9.1.4

7) Find a formula \(\displaystyle a_n\) for the \(\displaystyle nth\) term of the arithmetic sequence whose first term is \(\displaystyle a_1=−3\) such that \(\displaystyle a_{n−1}−a_n=4\) for \(\displaystyle n≥1\).

Solution: \(\displaystyle a_n=4n−7\)

8) Find a formula \(\displaystyle a_n\) for the \(\displaystyle nth\) term of the geometric sequence whose first term is \(\displaystyle a_1=1\) such that \(\displaystyle \frac{a_{n+1}}{a_n}=10\) for \(\displaystyle n≥1\).

9) Find a formula \(\displaystyle a_n\) for the \(\displaystyle nth\) term of the geometric sequence whose first term is \(\displaystyle a_1=3\) such that \(\displaystyle \frac{a_{n+1}}{a_n}=1/10\) for \(\displaystyle n≥1\).

Solution: \(\displaystyle a_n=3.10^{1−n}=30.10^{−n}\)

10) Find an explicit formula for the \(\displaystyle nth\) term of the sequence whose first several terms are \(\displaystyle {0,3,8,15,24,35,48,63,80,99,…}.\) (Hint: First add one to each term.)

11) Find an explicit formula for the \(\displaystyle nth\) term of the sequence satisfying \(\displaystyle a_1=0\) and \(\displaystyle a_n=2a_{n−1}+1\) for \(\displaystyle n≥2\).

Solution: \(\displaystyle a_n=2^n−1\)

 

Find a formula for the general term \(\displaystyle a_n\) of each of the following sequences.

12) \(\displaystyle {1,0,−1,0,1,0,−1,0,…}\) (Hint: Find where \(\displaystyle sinx\) takes these values)

13) \(\displaystyle {1,−1/3,1/5,−1/7,…}\)

Solution: \(\displaystyle a_n=\frac{(−1)^{n−1}}{2n−1}\)

 

Find a function \(\displaystyle f(n)\) that identifies the \(\displaystyle nth\) term \(\displaystyle a_n\) of the following recursively defined sequences, as \(\displaystyle a_n=f(n)\).

14) \(\displaystyle a_1=1\) and \(\displaystyle a_{n+1}=−a_n\) for \(\displaystyle n≥1\)

15) \(\displaystyle a_1=2\) and \(\displaystyle a_{n+1}=2a_n\) for \(\displaystyle n≥1\)

Solution: \(\displaystyle f(n)=2^n\)

16) \(\displaystyle a_1=1\) and \(\displaystyle a_{n+1}=(n+1)a_n\) for \(\displaystyle n≥1\)

17) \(\displaystyle a_1=2\) and \(\displaystyle a_{n+1}=(n+1)a_n/2\) for \(\displaystyle n≥1\)

Solution: \(\displaystyle a_1=1\) and \(\displaystyle a_{n+1}=a_n/2^n\) for \(\displaystyle n≥1\)

 

Plot the first \(\displaystyle N\) terms of each sequence. State whether the graphical evidence suggests that the sequence converges or diverges.

18) [T] \(\displaystyle a_1=1, a_2=2\), and for \(\displaystyle n≥2, a_n=\frac{1}{2}(a_{n−1}+a_{n−2})\); \(\displaystyle N=30\)

Solution: Terms oscillate above and below \(\displaystyle 5/3\) and appear to converge to \(\displaystyle 5/3\).

19) [T] \(\displaystyle a_1=1, a_2=2, a_3=3\) and for \(\displaystyle n≥4, a_n=\frac{1}{3}(a_{n−1}+a_{n−2}+a_{n−3}), N=30\)

20) [T] \(\displaystyle a_1=1, a_2=2\), and for \(\displaystyle n≥3, a_n=\sqrt{a_{n−1}a_{n−2}}; N=30\)

Solution: Terms oscillate above and below \(\displaystyle y≈1.57..\) and appear to converge to a limit.

21) [T] \(\displaystyle a_1=1, a_2=2, a_3=3\), and for \(\displaystyle n≥4, a_n=\sqrt{a_{n−1}a_{n−2}a_{n−3}}; N=30\)

 

Suppose that \(\displaystyle \lim_{n→∞}a_n=1, \lim_{n→∞}b_n=−1\), and \(\displaystyle 0<−b_n<a_n\) for all \(\displaystyle n\). Evaluate each of the following limits, or state that the limit does not exist, or state that there is not enough information to determine whether the limit exists.

22) \(\displaystyle \lim_{n→∞}3a_n−4b_n\)

Solution: \(\displaystyle 7\)

23) \(\displaystyle \lim_{n→∞}\frac{1}{2}b_n−\frac{1}{2}a_n\)

24) \(\displaystyle \lim_{n→∞}\frac{a_n+b_n}{a_n−b_n}\)

Solution: \(\displaystyle 0\)

25) \(\displaystyle \lim_{n→∞}\frac{a_n−b_n}{a_n+b_n}\)

 

Find the limit of each of the following sequences, using L’Hôpital’s rule when appropriate.

26) \(\displaystyle \frac{n^2}{2^n}\)

Solution: \(\displaystyle 0\)

27) \(\displaystyle \frac{(n−1)^2}{(n+1)^2}\)

28) \(\displaystyle \frac{\sqrt{n}}{\sqrt{n+1}}\)

Solution: \(\displaystyle 1\)

29) \(\displaystyle n^{1/n}\) (Hint: \(\displaystyle n^{1/n}=e^{\frac{1}{n}lnn})\)

 

For each of the following sequences, whose \(\displaystyle nth\) terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing.

30) \(\displaystyle n/2^n, n≥2\)

Solution: bounded, decreasing for \(\displaystyle n≥1\)

31) \(\displaystyle ln(1+\frac{1}{n})\)

32) \(\displaystyle sinn\)

Solution: bounded, not monotone

33) \(\displaystyle cos(n^2)\)

34) \(\displaystyle n^{1/n}, n≥3\)

Solution: bounded, decreasing

35) \(\displaystyle n^{−1/n}, n≥3\)

36) \(\displaystyle tann\)

Solution: not monotone, not bounded

 

Determine whether the sequence defined as follows has a limit. If it does, find the limit.

37) \(\displaystyle a_1=\sqrt{2}, a_2=\sqrt{2\sqrt{2}}. a_3=\sqrt{2\sqrt{2\sqrt{2}}}\) etc.

Determine whether the sequence defined as follows has a limit. If it does, find the limit.

38) \(\displaystyle a_1=3, a_n=\sqrt{2a_{n−1}}, n=2,3,….\)

Solution: \(\displaystyle a-n\) is decreasing and bounded below by \(\displaystyle 2\). The limit a must satisfy \(\displaystyle a=\sqrt{2a}\) so \(\displaystyle a=2\), independent of the initial value.

 

Use the Squeeze Theorem to find the limit of each of the following sequences.

39) \(\displaystyle nsin(1/n)\)

40) \(\displaystyle \frac{cos(1/n)−1}{1/n}\)

Solution: \(\displaystyle 0\)

41) \(\displaystyle a_n=\frac{n!}{n^n}\)

42) \(\displaystyle a_n=sinnsin(1/n)\)

Solution: \(\displaystyle 0:|sinx|≤|x|\) and \(\displaystyle |sinx|≤1\) so \(\displaystyle −\frac{1}{n}≤a_n≤\frac{1}{n})\).

 

For the following sequences, plot the first \(\displaystyle 25\) terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges.

43) [T] \(\displaystyle a_n=sinn\)

44) [T] \(\displaystyle a_n=cosn\)

Solution: Graph oscillates and suggests no limit.

Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit.

45) \(\displaystyle a_n=tan^{−1}(n^2)\)

46) \(\displaystyle a_n=(2n)^{1/n}−n^{1/n}\)

Solution: \(\displaystyle n^{1/n}→1\) and \(\displaystyle 2^{1/n}→1,\) so \(\displaystyle a_n→0\)

47) \(\displaystyle a_n=\frac{ln(n^2)}{ln(2n)}\)

48) \(\displaystyle a_n=(1−\frac{2}{n})^n\)

Solution: Since \(\displaystyle (1+1/n)^n→e\), one has \(\displaystyle (1−2/n)^n≈(1+k)^{−2k}→e^{−2}\) as \(\displaystyle k→∞.\)

49) \(\displaystyle a_n=ln(\frac{n+2}{n^2−3})\)

50) \(\displaystyle a_n=\frac{2^n+3^n}{4^n}\)

Solution: \(\displaystyle 2^n+3^n≤2⋅3^n\) and \(\displaystyle 3^n/4^n→0\) as \(\displaystyle n→∞\), so \(\displaystyle a_n→0\) as \(\displaystyle n→∞.\)

51) \(\displaystyle a_n=\frac{(1000)^n}{n!}\)

52) \(\displaystyle a_n=\frac{(n!)^2}{(2n)!}\)

Solution: \(\displaystyle \frac{a_{n+1}}{a_n}=n!/(n+1)(n+2)⋯(2n) =\frac{1⋅2⋅3⋯n}{(n+1)(n+2)⋯(2n)}<1/2^n\). In particular, \(\displaystyle a_{n+1}/a_n≤1/2\), so \(\displaystyle a_n→0\) as \(\displaystyle n→∞\).

 

Newton’s method seeks to approximate a solution \(\displaystyle f(x)=0\) that starts with an initial approximation \(\displaystyle x_0\) and successively defines a sequence \(\displaystyle x_{n+1}=x_n−\frac{f(x_n)}{f′(x_n)}\). For the given choice of \(\displaystyle f\) and \(\displaystyle x_0\), write out the formula for \(\displaystyle x_{n+1}\). If the sequence appears to converge, give an exact formula for the solution \(\displaystyle x\), then identify the limit \(\displaystyle x\) accurate to four decimal places and the smallest \(\displaystyle n\) such that \(\displaystyle x_n\) agrees with \(\displaystyle x\) up to four decimal places.

53) [T] \(\displaystyle f(x)=x^2−2, x_0=1\)

54) [T] \(\displaystyle f(x)=(x−1)^2−2, x_0=2\)

Solution: \(\displaystyle x_{n+1}=x_n−((x_n−1)^2−2)/2(x_n−1); x=1+\sqrt{2}, x≈2.4142, n=5\)

55) [T] \(\displaystyle f(x)=e^x−2, x_0=1\)

56) [T] \(\displaystyle f(x)=lnx−1, x_0=2\)

Solution: \(\displaystyle x_{n+1}=x_n−x_n(ln(x_n)−1); x=e, x≈2.7183, n=5\)

 

57) [T] Suppose you start with one liter of vinegar and repeatedly remove \(\displaystyle 0.1L\), replace with water, mix, and repeat.

a. Find a formula for the concentration after \(\displaystyle n\) steps.

b. After how many steps does the mixture contain less than \(\displaystyle 10%\) vinegar?

58) [T] A lake initially contains \(\displaystyle 2000\) fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by \(\displaystyle 6%\) each month. However, factoring in all causes, \(\displaystyle 150\) fish are lost each month.

a. Explain why the fish population after \(\displaystyle n\) months is modeled by \(\displaystyle P_n=1.06P_{n−1}−150\) with \(\displaystyle P_0=2000\).

b. How many fish will be in the pond after one year?

Solution: a. Without losses, the population would obey \(\displaystyle P_n=1.06P_{n−1}\). The subtraction of \(\displaystyle 150\) accounts for fish losses. b. After \(\displaystyle 12\) months, we have \(\displaystyle P_{12}≈1494.\)

59) [T] A bank account earns \(\displaystyle 5%\) interest compounded monthly. Suppose that \(\displaystyle $1000\) is initially deposited into the account, but that \(\displaystyle $10\) is withdrawn each month.

a. Show that the amount in the account after \(\displaystyle n\) months is \(\displaystyle A_n=(1+.05/12)A_{n−1}−10; A_0=1000.\)

b. How much money will be in the account after \(\displaystyle 1\) year?

c. Is the amount increasing or decreasing?

d. Suppose that instead of \(\displaystyle $10\), a fixed amount \(\displaystyle d\) dollars is withdrawn each month. Find a value of \(\displaystyle d\) such that the amount in the account after each month remains \(\displaystyle $1000\).

e. What happens if \(\displaystyle d\) is greater than this amount?

60) [T] A student takes out a college loan of \(\displaystyle $10,000\) at an annual percentage rate of \(\displaystyle 6%,\) compounded monthly.

a. If the student makes payments of \(\displaystyle $100\) per month, how much does the student owe after \(\displaystyle 12\) months?

b. After how many months will the loan be paid off?

Solution: a. The student owes \(\displaystyle $9383\) after \(\displaystyle 12\) months. b. The loan will be paid in full after \(\displaystyle 139\) months or eleven and a half years.

61) [T] Consider a series combining geometric growth and arithmetic decrease. Let \(\displaystyle a_1=1\). Fix \(\displaystyle a>1\) and \(\displaystyle 0<b<a\). Set \(\displaystyle a_{n+1}=a.a_n−b.\) Find a formula for \(\displaystyle a_{n+1}\) in terms of \(\displaystyle a_n, a\), and \(\displaystyle b\) and a relationship between \(\displaystyle a\) and \(\displaystyle b\) such that \(\displaystyle a_n\) converges.

62) [T] The binary representation \(\displaystyle x=0.b_1b_2b_3...\) of a number \(\displaystyle x\) between \(\displaystyle 0\) and \(\displaystyle 1\) can be defined as follows. Let \(\displaystyle b_1=0\) if \(\displaystyle x<1/2\) and \(\displaystyle b_1=1\) if \(\displaystyle 1/2≤x<1.\) Let \(\displaystyle x_1=2x−b_1\). Let \(\displaystyle b_2=0\) if \(\displaystyle x_1<1/2\) and \(\displaystyle b_2=1\) if \(\displaystyle 1/2≤x<1\). Let \(\displaystyle x_2=2x_1−b_2\) and in general, \(\displaystyle x_n=2x_{n−1}−b_n\) and \(\displaystyle b_{n−}1=0\) if \(\displaystyle x_n<1/2\) and \(\displaystyle b_{n−1}=1\) if \(\displaystyle 1/2≤x_n<1\). Find the binary expansion of \(\displaystyle 1/3\).

Solution: \(\displaystyle b_1=0, x_1=2/3, b_2=1, x_2=4/3−1=1/3,\) so the pattern repeats, and \(\displaystyle 1/3=0.010101….\)

63) [T] To find an approximation for \(\displaystyle π\), set \(\displaystyle a_0=\sqrt{2+1}, a_1=\sqrt{2+a_0}\), and, in general, \(\displaystyle a_{n+1}=\sqrt{2+a_n}\). Finally, set \(\displaystyle p_n=3.2^n\sqrt{2−a_n}\). Find the first ten terms of \(\displaystyle p_n\) and compare the values to \(\displaystyle π\).

 

For the following two exercises, assume that you have access to a computer program or Internet source that can generate a list of zeros and ones of any desired length. Pseudorandom number generators (PRNGs) play an important role in simulating random noise in physical systems by creating sequences of zeros and ones that appear like the result of flipping a coin repeatedly. One of the simplest types of PRNGs recursively defines a random-looking sequence of \(\displaystyle N\) integers \(\displaystyle a_1,a_2,…,a_N\) by fixing two special integers \(\displaystyle (K\) and \(\displaystyle M\) and letting \(\displaystyle a_{n+1}\) be the remainder after dividing \(\displaystyle K.a_n\) into \(\displaystyle M\), then creates a bit sequence of zeros and ones whose \(\displaystyle nth\) term \(\displaystyle b_n\) is equal to one if \(\displaystyle a_n\) is odd and equal to zero if \(\displaystyle a_n\) is even. If the bits \(\displaystyle b_n\) are pseudorandom, then the behavior of their average \(\displaystyle (b_1+b_2+⋯+b_N)/N\) should be similar to behavior of averages of truly randomly generated bits.

64) [T] Starting with \(\displaystyle K=16,807\) and \(\displaystyle M=2,147,483,647\), using ten different starting values of \(\displaystyle a_1\), compute sequences of bits \(\displaystyle b_n\) up to \(\displaystyle n=1000,\) and compare their averages to ten such sequences generated by a random bit generator.

Solution: For the starting values \(\displaystyle a_1=1, a_2=2,…, a_1=10,\) the corresponding bit averages calculated by the method indicated are \(\displaystyle 0.5220, 0.5000, 0.4960, 0.4870, 0.4860, 0.4680, 0.5130, 0.5210, 0.5040,\) and \(\displaystyle 0.4840\). Here is an example of ten corresponding averages of strings of \(\displaystyle 1000\) bits generated by a random number generator: \(\displaystyle 0.4880, 0.4870, 0.5150, 0.5490, 0.5130, 0.5180, 0.4860, 0.5030, 0.5050, 0.4980.\) There is no real pattern in either type of average. The random-number-generated averages range between \(\displaystyle 0.4860\) and \(\displaystyle 0.5490\), a range of \(\displaystyle 0.0630\), whereas the calculated PRNG bit averages range between \(\displaystyle 0.4680\) and \(\displaystyle 0.5220\), a range of \(\displaystyle 0.0540.\)

65) [T] Find the first \(\displaystyle 1000\) digits of \(\displaystyle π\) using either a computer program or Internet resource. Create a bit sequence \(\displaystyle b_n\) by letting \(\displaystyle b_n=1\) if the \(\displaystyle nth\) digit of \(\displaystyle π\) is odd and \(\displaystyle b_n=0\) if the \(\displaystyle nth\) digit of \(\displaystyle π\) is even. Compute the average value of \(\displaystyle b_n\) and the average value of \(\displaystyle d_n=|b_{n+1}−b_n|, n=1,...,999.\) Does the sequence \(\displaystyle b_n\) appear random? Do the differences between successive elements of \(\displaystyle b_n\) appear random?

9.2: Infinite Series

Using sigma notation, write the following expressions as infinite series.

1) \(\displaystyle 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+⋯\)

Solution: \(\displaystyle \sum_{n=1}^∞\frac{1}{n}\)

2) \(\displaystyle 1−1+1−1+⋯\)

3) \(\displaystyle 1−\frac{1}{2}+\frac{1}{3}−\frac{1}{4}+...\)

Solution: \(\displaystyle \sum_{n=1}^∞\frac{(−1)^{n−1}}{n}\)

4) \(\displaystyle sin1+sin1/2+sin1/3+sin1/4+⋯\)

 

Compute the first four partial sums \(\displaystyle S_1,…,S_4\) for the series having \(\displaystyle nth\) term \(\displaystyle a_n\) starting with \(\displaystyle n=1\) as follows.

5) \(\displaystyle a_n=n\)

Solution: \(\displaystyle 1,3,6,10\)

6) \(\displaystyle a_n=1/n\)

7) \(\displaystyle a_n=sin(nπ/2)\)

Solution: \(\displaystyle 1,1,0,0\)

8) \(\displaystyle a_n=(−1)^n\)

 

In the following exercises, compute the general term \(\displaystyle a_n\) of the series with the given partial sum \(\displaystyle S_n\). If the sequence of partial sums converges, find its limit \(\displaystyle S\).

9) \(\displaystyle S_n=1−\frac{1}{n}, n≥2\)

Solution: \(\displaystyle a_n=S_n−S_{n−1}=\frac{1}{n−1}−\frac{1}{n}.\) Series converges to \(\displaystyle S=1.\)

10) \(\displaystyle S_n=\frac{n(n+1)}{2}, n≥1\)

11) \(\displaystyle S_n=\sqrt{n},n≥2\)

Solution: \(\displaystyle a_n=S_n−S_{n−1}=\sqrt{n}−\sqrt{n−1}=\frac{1}{\sqrt{n−1}+\sqrt{n}}.\) Series diverges because partial sums are unbounded.

12) \(\displaystyle S_n=2−(n+2)/2^n,n≥1\)

 

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges.

13) \(\displaystyle \sum_{n=1}^∞\frac{n}{n+2}\)

Solution: \(\displaystyle S_1=1/3, S_2=1/3+2/4>1/3+1/3=2/3, S_3=1/3+2/4+3/5>3⋅(1/3)=1.\) In general \(\displaystyle S_k>k/3.\) Series diverges.

14) \(\displaystyle \sum_{n=1}^∞(1−(−1)^n))\)

15) \(\displaystyle \sum_{n=1}^∞\frac{1}{(n+1)(n+2)}\) (Hint: Use a partial fraction decomposition like that for \(\displaystyle \sum_{n=1}^∞\frac{1}{n(n+1)}.)\)

Solution:

\(\displaystyle S_1=1/(2.3)=1/6=2/3−1/2,\)

\(\displaystyle S_2=1/(2.3)+1/(3.4)=2/12+1/12=1/4=3/4−1/2,\)

\(\displaystyle S_3=1/(2.3)+1/(3.4)+1/(4.5)=10/60+5/60+3/60=3/10=4/5−1/2,\)

\(\displaystyle S_4=1/(2.3)+1/(3.4)+1/(4.5)+1/(5.6)=10/60+5/60+3/60+2/60=1/3=5/6−1/2.\)

The pattern is \(\displaystyle S_k=(k+1)/(k+2)−1/2\) and the series converges to \(\displaystyle 1/2.\)

16) \(\displaystyle \sum_{n=1}^∞\frac{1}{2n+1}\) (Hint: Follow the reasoning for \(\displaystyle \sum_{n=1}^∞\frac{1}{n}.)\)

 

Suppose that \(\displaystyle \sum_{n=1}^∞a_n=1\), that \(\displaystyle \sum_{n=1}^∞b_n=−1\), that \(\displaystyle a_1=2\), and \(\displaystyle b_1=−3\). Find the sum of the indicated series.

17) \(\displaystyle \sum_{n=1}^∞(a_n+b_n)\)

Solution: \(\displaystyle 0\)

18) \(\displaystyle \sum_{n=1}^∞(a_n−2b_n)\)

19) \(\displaystyle \sum_{n=2}^∞(a_n−b_n)\)

Solution: \(\displaystyle −3\)

20) \(\displaystyle \sum_{n=1}^∞(3a_{n+1}−4b_{n+1})\)

 

State whether the given series converges and explain why.

21) \(\displaystyle \sum_{n=1}^∞\frac{1}{n+1000}\) (Hint: Rewrite using a change of index.)

Solution: diverges, \(\displaystyle \sum_{n=1001}^∞\frac{1}{n}\)

22) \(\displaystyle \sum_{n=1}^∞\frac{1}{n+10^{80}}\) (Hint: Rewrite using a change of index.)

23) \(\displaystyle 1+\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+⋯\)

Solution: convergent geometric series, \(\displaystyle r=1/10<1\)

24) \(\displaystyle 1+\frac{e}{π}+\frac{e^2}{π^2}+\frac{e^3}{π^3}+⋯\)

25) \(\displaystyle 1+\frac{π}{e}+\frac{π^2}{e^4}+\frac{π^3}{e^6}+\frac{π^4}{e^8}+⋯\)

Solution: convergent geometric series, \(\displaystyle r=π/e^2<1\)

26) \(\displaystyle 1−\sqrt{\frac{π}{3}}+\sqrt{\frac{π^2}{9}}−\sqrt{\frac{π^3}{27}}+⋯\)

 

For \(\displaystyle a_n\) as follows, write the sum as a geometric series of the form \(\displaystyle \sum_{n=1}^∞ar^n\). State whether the series converges and if it does, find the value of \(\displaystyle \sum a_n\).

27) \(\displaystyle a_1=−1\) and \(\displaystyle a_n/a_{n+1}=−5\) for \(\displaystyle n≥1.\)

Solution: \(\displaystyle \sum_{n=1}^∞5⋅(−1/5)^n\), converges to \(\displaystyle −5/6\)

28) \(\displaystyle a_1=2\) and \(\displaystyle a_n/a_{n+1}=1/2\) for \(\displaystyle n≥1.\)

29) \(\displaystyle a_1=10\) and \(\displaystyle a_n/a_{n+1}=10\) for \(\displaystyle n≥1\).

Solution: \(\displaystyle \sum_{n=1}^∞100⋅(1/10)^n,\) converges to \(\displaystyle 100/9\)

30) \(\displaystyle a_1=1/10\) and \(\displaystyle a_n/a_{n+1}=−10\) for \(\displaystyle n≥1\).

 

Use the identity \(\displaystyle \frac{1}{1−y}=\sum_{n=0}^∞y^n\) to express the function as a geometric series in the indicated term.

31) \(\displaystyle \frac{x}{1+x}\) in \(\displaystyle x\)

Solution: \(\displaystyle x\sum_{n=0}^∞(−x)^n=\sum_{n=1}^∞(−1)^{n−1}x^n\)

32) \(\displaystyle \frac{\sqrt{x}}{1−x^{3/2}}\) in \(\displaystyle \sqrt{x}\)

33) \(\displaystyle \frac{1}{1+sin^2x}\) in \(\displaystyle sinx\)

Solution: \(\displaystyle \sum_{n=0}^∞(−1)^nsin^2n(x)\)

34) \(\displaystyle sec^2x\) in \(\displaystyle sinx\)

 

Evaluate the following telescoping series or state whether the series diverges.

35) \(\displaystyle \sum_{n=1}^∞2^{1/n}−2^{1/(n+1)}\)

Solution: \(\displaystyle S_k=2−2^{1/(k+1)}→1\) as \(\displaystyle k→∞.\)

36) \(\displaystyle \sum_{n=1}^∞\frac{1}{n^{13}}−\frac{1}{(n+1)^{13}}\)

37) \(\displaystyle \sum_{n=1}^∞(\sqrt{n}−\sqrt{n+1})\)

Soution: \(\displaystyle S_k=1−\sqrt{k+1}\) diverges

38) \(\displaystyle \sum_{n=1}^∞(sinn−sin(n+1))\)

 

Express the following series as a telescoping sum and evaluate its nth partial sum.

39) \(\displaystyle \sum_{n=1}^∞ln(\frac{n}{n+1})\)

Solution: \(\displaystyle \sum_{n=1}^∞lnn−ln(n+1),S_k=−ln(k+1)\)

40) \(\displaystyle \sum_{n=1}^∞\frac{2n+1}{(n^2+n)^2}\) (Hint: Factor denominator and use partial fractions.)

41) \(\displaystyle \sum_{n=2}^∞\frac{ln(1+\frac{1}{n})}{lnnln(n+1)}\)

Solution: \(\displaystyle a_n=\frac{1}{lnn}−\frac{1}{ln(n+1)}\) and \(\displaystyle S_k=\frac{1}{ln(2)}−\frac{1}{ln(k+1)}→\frac{1}{ln(2)}\)

42) \(\displaystyle \sum_{n=1}^∞\frac{(n+2)}{n(n+1)2^{n+1}}\) (Hint: Look at \(\displaystyle 1/(n2^n)\).

 

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.

43) Let \(\displaystyle a_n=f(n)−2f(n+1)+f(n+2),\) in which \(\displaystyle f(n)→0\) as \(\displaystyle n→∞.\) Find \(\displaystyle \sum_{n=1}^∞a_n\).

Solution: \(\displaystyle sum_{n=1}^∞a_n=f(1)−f(2)\)

44) \(\displaystyle a_n=f(n)−f(n+1)−f(n+2)+f(n+3),\) in which \(\displaystyle f(n)→0\) as \(\displaystyle n→∞\). Find \(\displaystyle \sum_{n=1}^∞a_n\).

45) Suppose that \(\displaystyle a_n=c_0f(n)+c_1f(n+1)+c_2f(n+2)+c_3f(n+3)+c_4f(n+4),\) where \(\displaystyle f(n)→0\) as \(\displaystyle n→∞\). Find a condition on the coefficients \(\displaystyle c_0,…,c_4\) that make this a general telescoping series.

Solution: \(\displaystyle c_0+c_1+c_2+c_3+c_4=0\)

46) Evaluate \(\displaystyle \sum_{n=1}^∞\frac{1}{n(n+1)(n+2)}\) (Hint: \(\displaystyle \frac{1}{n(n+1)(n+2)}=\frac{1}{2n}−\frac{1}{n+1}+\frac{1}{2(n+2)})\)

47) Evaluate \(\displaystyle \sum_{n=2}^∞\frac{2}{n^3−n}.\)

Solution: \(\displaystyle \frac{2}{n^3−1}=\frac{1}{n−1}−\frac{2}{n}+\frac{1}{n+1}, S_n=(1−1+1/3)+(1/2−2/3+1/4) +(1/3−2/4+1/5)+(1/4−2/5+1/6)+⋯=1/2\)

48) Find a formula for \(\displaystyle \sum_{n=1}^∞\frac{1}{n(n+N)}\) where \(\displaystyle N\) is a positive integer.

49) [T] Define a sequence \(\displaystyle t_k=\sum_{n=1}^{k−1}(1/k)−lnk\). Use the graph of \(\displaystyle 1/x\) to verify that \(\displaystyle t_k\) is increasing. Plot \(\displaystyle t_k\) for \(\displaystyle k=1…100\) and state whether it appears that the sequence converges.

Solution: \(\displaystyle t_k\) converges to \(\displaystyle 0.57721…t_k\) is a sum of rectangles of height \(\displaystyle 1/k\) over the interval \(\displaystyle [k,k+1]\) which lie above the graph of \(\displaystyle 1/x\).

50) [T] Suppose that \(\displaystyle N\) equal uniform rectangular blocks are stacked one on top of the other, allowing for some overhang. Archimedes’ law of the lever implies that the stack of \(\displaystyle N\) blocks is stable as long as the center of mass of the top \(\displaystyle (N−1)\) blocks lies at the edge of the bottom block. Let \(\displaystyle x\) denote the position of the edge of the bottom block, and think of its position as relative to the center of the next-to-bottom block. This implies that \(\displaystyle (N−1)x=(\frac{1}{2}−x)\) or \(\displaystyle x=1/(2N)\). Use this expression to compute the maximum overhang (the position of the edge of the top block over the edge of the bottom block.) See the following figure.

Each of the following infinite series converges to the given multiple of \(\displaystyle π\) or \(\displaystyle 1/π\).

 

In each case, find the minimum value of \(\displaystyle N\) such that the \(\displaystyle Nth\) partial sum of the series accurately approximates the left-hand side to the given number of decimal places, and give the desired approximate value. Up to \(\displaystyle 15\) decimals place, \(\displaystyle π=3.141592653589793....\)

51) [T] \(\displaystyle π=−3+\sum_{n=1}^∞\frac{n2^nn!^2}{(2n)!},\) error \(\displaystyle <0.0001\)

Solution: \(\displaystyle N=22, S_N=6.1415\)

52) [T] \(\displaystyle \frac{π}{2}=\sum_{k=0}^∞\frac{k!}{(2k+1)!!}=\sum_{k=0}^∞\frac{2^kk!^2}{(2k+1)!},\) error \(\displaystyle <10^{−4}\)

53) [T] \(\displaystyle \frac{9801}{2π}=\frac{4}{9801}\sum_{k=0}^∞\frac{(4k)!(1103+26390k)}{(k!)^4396^{4k}},\) error \(\displaystyle <10^{−12}\)

Solution: \(\displaystyle N=3, S_N=1.559877597243667...\)

54) [T] \(\displaystyle \frac{1}{12π}=\sum_{k=0}^∞\frac{(−1)^k(6k)!(13591409+545140134k)}{(3k)!(k!)^3640320^{3k+3/2}}\), error \(\displaystyle <10^{−15}\)

55) [T] A fair coin is one that has probability \(\displaystyle 1/2\) of coming up heads when flipped.

a. What is the probability that a fair coin will come up tails \(\displaystyle n\) times in a row?

b. Find the probability that a coin comes up heads for the first time after an even number of coin flips.

Solution: a. The probability of any given ordered sequence of outcomes for \(\displaystyle n\) coin flips is \(\displaystyle 1/2^n\). b. The probability of coming up heads for the first time on the \(\displaystyle n\) th flip is the probability of the sequence \(\displaystyle TT…TH\) which is \(\displaystyle 1/2^n\). The probability of coming up heads for the first time on an even flip is \(\displaystyle \sum_{n=1}^∞1/2^{2n}\) or \(\displaystyle 1/3\).

56) [T] Find the probability that a fair coin is flipped a multiple of three times before coming up heads.

57) [T] Find the probability that a fair coin will come up heads for the second time after an even number of flips.

Solution: \(\displaystyle 5/9\)

58) [T] Find a series that expresses the probability that a fair coin will come up heads for the second time on a multiple of three flips.

59) [T] The expected number of times that a fair coin will come up heads is defined as the sum over \(\displaystyle n=1,2,…\) of \(\displaystyle n\) times the probability that the coin will come up heads exactly \(\displaystyle n\) times in a row, or \(\displaystyle n/2^{n+1}\). Compute the expected number of consecutive times that a fair coin will come up heads.

Solution: \(\displaystyle E=\sum_{n=1}^∞n/2^{n+1}=1,\) as can be shown using summation by parts

60) [T] A person deposits \(\displaystyle $10\) at the beginning of each quarter into a bank account that earns \(\displaystyle 4%\) annual interest compounded quarterly (four times a year).

a. Show that the interest accumulated after \(\displaystyle n\) quarters is \(\displaystyle $10(\frac{1.01^{n+1}−1}{0.01}−n).\)

b. Find the first eight terms of the sequence.

c. How much interest has accumulated after \(\displaystyle 2\) years?

61) [T] Suppose that the amount of a drug in a patient’s system diminishes by a multiplicative factor \(\displaystyle r<1\) each hour. Suppose that a new dose is administered every \(\displaystyle N\) hours. Find an expression that gives the amount \(\displaystyle A(n)\) in the patient’s system after \(\displaystyle n\) hours for each \(\displaystyle n\) in terms of the dosage \(\displaystyle d\) and the ratio \(\displaystyle r\). (Hint: Write \(\displaystyle n=mN+k\), where \(\displaystyle 0≤k<N\), and sum over values from the different doses administered.)

Solution: The part of the first dose after \(\displaystyle n\) hours is \(\displaystyle dr^n\), the part of the second dose is \(\displaystyle dr^{n−N}\), and, in general, the part remaining of the \(\displaystyle mth\) dose is \(\displaystyle dr^{n−mN}\), so \(\displaystyle A(n)=\sum_{l=0}^mdr^{n−lN}=\sum_{l=0}^mdr^{k+(m−l)N}=\sum_{q=0}^mdr^{k+qN}=dr^k\sum_{q=0}^mr^{Nq}=dr^k\frac{1−r^{(m+1)N}}{1−r^N},n=k+mN.\)

62) [T] A certain drug is effective for an average patient only if there is at least \(\displaystyle 1\) mg per kg in the patient’s system, while it is safe only if there is at most \(\displaystyle 2\) mg per kg in an average patient’s system. Suppose that the amount in a patient’s system diminishes by a multiplicative factor of \(\displaystyle 0.9\) each hour after a dose is administered. Find the maximum interval \(\displaystyle N\) of hours between doses, and corresponding dose range \(\displaystyle d\) (in mg/kg) for this \(\displaystyle N\) that will enable use of the drug to be both safe and effective in the long term.

63) Suppose that \(\displaystyle a_n≥0\) is a sequence of numbers. Explain why the sequence of partial sums of \(\displaystyle a_n\) is increasing.

Solution: \(\displaystyle S_{N+1}=a_{N+1}+S_N≥S_N\)

64) [T] Suppose that \(\displaystyle a_n\) is a sequence of positive numbers and the sequence \(\displaystyle S_n\) of partial sums of \(\displaystyle a_n\) is bounded above. Explain why \(\displaystyle \sum_{n=1}^∞a_n\) converges. Does the conclusion remain true if we remove the hypothesis \(\displaystyle a_n≥0\)?

65) [T] Suppose that \(\displaystyle a_1=S_1=1\) and that, for given numbers \(\displaystyle S>1\) and \(\displaystyle 0<k<1\), one defines \(\displaystyle a_{n+1}=k(S−S_n)\) and \(\displaystyle S_{n+1}=a_{n+1}+S_n\). Does \(\displaystyle S_n\) converge? If so, to what? (Hint: First argue that \(\displaystyle S_n<S\) for all \(\displaystyle n\) and \(\displaystyle S_n\) is increasing.)

Solution: Since \(\displaystyle S>1, a_2>0,\) and since \(\displaystyle k<1, S_2=1+a_2<1+(S−1)=S\). If \(\displaystyle S_n>S\) for some \(\displaystyle n\), then there is a smallest \(\displaystyle n\). For this \(\displaystyle n, S>S_{n−1}\), so \(\displaystyle S_n=S_{n−1}+k(S−S_{n−1})=kS+(1−k)S_{n−1}<S\), a contradiction. Thus \(\displaystyle S_n<S\) and \(\displaystyle a_{n+1}>0\) for all \(\displaystyle n\), so \(\displaystyle S_n\) is increasing and bounded by \(\displaystyle S\). Let \(\displaystyle S_∗=\lim S_n\). If \(\displaystyle S_∗<S\), then \(\displaystyle δ=k(S−S_∗)>0\), but we can find n such that \(\displaystyle S_∗−S_n<δ/2\), which implies that \(\displaystyle S_{n+1}=S_n+k(S−S_n) >S_∗+δ/2\), contradicting that Sn is increasing to \(\displaystyle S_∗\). Thus \(\displaystyle S_n→S.\)

66) [T] A version of von Bertalanffy growth can be used to estimate the age of an individual in a homogeneous species from its length if the annual increase in year \(\displaystyle n+1\) satisfies \(\displaystyle a_{n+1}=k(S−S_n)\), with \(\displaystyle S_n\) as the length at year \(\displaystyle n, S\) as a limiting length, and \(\displaystyle k\) as a relative growth constant. If \(\displaystyle S_1=3, S=9,\) and \(\displaystyle k=1/2,\) numerically estimate the smallest value of n such that \(\displaystyle S_n≥8\). Note that \(\displaystyle S_{n+1}=S_n+a_{n+1}.\) Find the corresponding \(\displaystyle n\) when \(\displaystyle k=1/4.\)

67) [T] Suppose that \(\displaystyle \sum_{n=1}^∞a_n\) is a convergent series of positive terms. Explain why \(\displaystyle \lim_{N→∞}\sum_{n=N+1}^∞a_n=0.\)

Solution: Let \(\displaystyle S_k=\sum_{n=1}^ka_n\) and \(\displaystyle S_k→L\). Then \(\displaystyle S_k\) eventually becomes arbitrarily close to \(\displaystyle L\), which means that \(\displaystyle L−S_N=\sum_{n=N+1}^∞a_n\) becomes arbitrarily small as \(\displaystyle N→∞.\)

68) [T] Find the length of the dashed zig-zag path in the following figure.

69) [T] Find the total length of the dashed path in the following figure.

Solution: \(\displaystyle L=(1+\frac{1}{2})\sum_{n=1}^∞1/2^n=\frac{3}{2}\).

70) [T] The Sierpinski triangle is obtained from a triangle by deleting the middle fourth as indicated in the first step, by deleting the middle fourths of the remaining three congruent triangles in the second step, and in general deleting the middle fourths of the remaining triangles in each successive step. Assuming that the original triangle is shown in the figure, find the areas of the remaining parts of the original triangle after \(\displaystyle N\) steps and find the total length of all of the boundary triangles after \(\displaystyle N\) steps.

71) [T] The Sierpinski gasket is obtained by dividing the unit square into nine equal sub-squares, removing the middle square, then doing the same at each stage to the remaining sub-squares. The figure shows the remaining set after four iterations. Compute the total area removed after \(\displaystyle N\) stages, and compute the length the total perimeter of the remaining set after \(\displaystyle N\) stages.

Solution: At stage one a square of area \(\displaystyle 1/9\) is removed, at stage \(\displaystyle 2\) one removes \(\displaystyle 8\) squares of area \(\displaystyle 1/9^2\), at stage three one removes \(\displaystyle 8^2\) squares of area \(\displaystyle 1/9^3\), and so on. The total removed area after \(\displaystyle N\) stages is \(\displaystyle \sum_{n=0}^{N−1}8^N/9^{N+1}=\frac{1}{8}(1−(8/9)^N)/(1−8/9)→1\) as \(\displaystyle N→∞.\) The total perimeter is \(\displaystyle 4+4\sum_{n=0}8^N/3^{N+1}→∞.\)

9.3: The Divergence and Integral Tests

For each of the following sequences, if the divergence test applies, either state that \(\displaystyle \lim_{n→∞}a_n\) does not exist or find \(\displaystyle \lim_{n→∞}a_n\). If the divergence test does not apply, state why.

1) \(\displaystyle a_n=\frac{n}{n+2}\)

2) \(\displaystyle a_n=\frac{n}{5n^2−3}\)

Solution: \(\displaystyle \lim_{n→∞}a_n=0\). Divergence test does not apply.

3) \(\displaystyle a_n=\frac{n}{\sqrt{3n^2+2n+1}}\)

4) \(\displaystyle a_n=\frac{(2n+1)(n−1)}{(n+1)^2}\)

Solution: \(\displaystyle \lim_{n→∞}a_n=2\). Series diverges.

5) \(\displaystyle a_n=\frac{(2n+1)^{2n}}{(3n2+1)^n}\)

6) \(\displaystyle a_n=\frac{2^n}{3^{n/2}}\)

Solution: \(\displaystyle \lim_{n→∞}a_n=∞\) (does not exist). Series diverges.

7) \(\displaystyle a_n=\frac{2^n+3^n}{10^{n/2}}\)

8) \(\displaystyle a_n=e^{−2/n}\)

Solution: \(\displaystyle \lim_{n→∞}a_n=1.\) Series diverges.

9) \(\displaystyle a_n=cosn\)

10) \(\displaystyle a_n=tann\)

Solution: \(\displaystyle \lim_{n→∞}a_n\) does not exist. Series diverges.

11) \(\displaystyle a_n=\frac{1−cos^2(1/n)}{sin^2(2/n)}\)

12) \(\displaystyle a_n=(1−\frac{1}{n})^{2n})\)

Solution: \(\displaystyle \lim_{n→∞}a_n=1/e^2.\) Series diverges.

13) \(\displaystyle a_n=\frac{lnn}{n}\)

14) \(\displaystyle a_n=\frac{(lnn)^2}{\sqrt{n}}\)

Solution: \(\displaystyle \lim_{n→∞}a_n=0.\) Divergence test does not apply.

 

State whether the given \(\displaystyle p\)-series converges.

15) \(\displaystyle \sum_{n=1}^∞\frac{1}{\sqrt{n}}\)

16) \(\displaystyle \sum_{n=1}^∞\frac{1}{n\sqrt{n}}\)

Solution: Series converges, \(\displaystyle p>1\).

17) \(\displaystyle \sum_{n=1}^∞\frac{1}{\sqrt[3]{n^2}}\)

18) \(\displaystyle \sum_{n=1}^∞\frac{1}{\sqrt[3]{n^4}}\)

Solution: Series converges, \(\displaystyle p=4/3>1.\)

19) \(\displaystyle \sum_{n=1}^∞\frac{n^e}{n^π}\)

20) \(\displaystyle \sum_{n=1}^∞\frac{n^π}{n^{2e}}\)

Solution: Series converges, \(\displaystyle p=2e−π>1.\)

 

Use the integral test to determine whether the following sums converge.

21) \(\displaystyle \sum_{n=1}^∞\frac{1}{\sqrt{n+5}}\)

22) \(\displaystyle \sum_{n=1}^∞\frac{1}{\sqrt[3]{n+5}}\)

Solution: Series diverges by comparison with \(\displaystyle ∫^∞_1\frac{dx}{(x+5)^{1/3}}\).

23) \(\displaystyle \sum_{n=2}^∞\frac{1}{nlnn}\)

24) \(\displaystyle \sum_{n=1}^∞\frac{n}{1+n^2}\)

Solution: Series diverges by comparison with \(\displaystyle ∫^∞_1\frac{x}{1+x^2}dx.\)

25) \(\displaystyle \sum_{n=1}^∞\frac{e^n}{1+e^{2n}}\)

26) \(\displaystyle \sum_{n=1}^∞\frac{2n}{1+n^4}\)

Solution: Series converges by comparison with \(\displaystyle ∫^∞_1\frac{2x}{1+x^4}dx.\)

27) \(\displaystyle \sum_{n=2}^∞\frac{1}{nln^2n}\)

 

Express the following sums as \(\displaystyle p\)-series and determine whether each converges.

28) \(\displaystyle \sum_{n=1}^∞2^{−lnn}\) (Hint: \(\displaystyle 2^{−lnn}=1/n^{ln2}.)\)

Solution: \(\displaystyle 2^{−lnn}=1/n^{ln2}.\) Since \(\displaystyle ln2<1\), diverges by \(\displaystyle p\)-series.

29) \(\displaystyle \sum_{n=1}^∞3^{−lnn}\) (Hint: \(\displaystyle 3^{−lnn}=1/n^{ln3}.)\)

30) \(\displaystyle \sum_{n=1}^n2^{−2lnn}\)

Solution: \(\displaystyle 2^{−2lnn}=1/n^{2ln2}.\) Since \(\displaystyle 2ln2−1<1\), diverges by \(\displaystyle p\)-series.

31) \(\displaystyle \sum_{n=1}^∞n3^{−2lnn}\)

 

Use the estimate \(\displaystyle R_N≤∫^∞_Nf(t)dt\) to find a bound for the remainder \(\displaystyle R_N=\sum_{n=1}^∞a_n−\sum_{n=1}^Na_n\) where \(\displaystyle a_n=f(n).\)

32) \(\displaystyle \sum_{n=1}^{1000}\frac{1}{n^2}\)

Solution: \(\displaystyle R_{1000}≤∫^∞_{1000}\frac{dt}{t^2}=−\frac{1}{t}∣^∞_{1000}=0.001\)

33) \(\displaystyle \sum_{n=1}^{1000}\frac{1}{n^3}\)

34) \(\displaystyle \sum_{n=1}^{1000}\frac{1}{1+n^2}\)

Solution: \(\displaystyle R_{1000}≤∫^∞_{1000}\frac{dt}{1+t^2}=tan^{−1}∞−tan^{−1}(1000)=π/2−tan^{−1}(1000)≈0.000999\)

35) \(\displaystyle \sum_{n=1}^{100}n/2^n\)

 

[T] Find the minimum value of \(\displaystyle N\) such that the remainder estimate \(\displaystyle ∫^∞_{N+1}f<R_N<∫^∞_Nf\) guarantees that \(\displaystyle \sum_{n=1}^Na_n\) estimates \(\displaystyle \sum_{n=1}^∞a_n,\) accurate to within the given error.

36) \(\displaystyle a_n=\frac{1}{n^2},\) error \(\displaystyle <10^{−4}\)

Solution: \(\displaystyle R_N<∫^∞_N\frac{dx}{x^2}=1/N,N>10^4\)

37) \(\displaystyle a_n=\frac{1}{n^{1.1}},\) error \(\displaystyle <10^{−4}\)

38) \(\displaystyle a_n=\frac{1}{n^{1.01}},\) error \(\displaystyle <10^{−4}\)

Solution: \(\displaystyle R_N<∫^∞_N\frac{dx}{x^{1.01}}=100N^{−0.01},N>10^{600}\)

39) \(\displaystyle a_n=\frac{1}{nln^2n},\) error \(\displaystyle <10^{−3}\)

40) \(\displaystyle a_n=\frac{1}{1+n^2},\) error \(\displaystyle <10^{−3}\)

Solution: \(\displaystyle R_N<∫^∞_N\frac{dx}{1+x^2}=π/2−tan^{−1}(N),N>tan(π/2−10^{−3})≈1000\)

 

In the following exercises, find a value of \(\displaystyle N\) such that \(\displaystyle R_N\) is smaller than the desired error. Compute the corresponding sum \(\displaystyle \sum_{n=1}^Na_n\) and compare it to the given estimate of the infinite series.

41) \(\displaystyle a_n=\frac{1}{n^{11}},\) error \(\displaystyle <10^{−4}, \sum_{n=1}^∞\frac{1}{n^{11}}=1.000494…\)

42) \(\displaystyle a_n=\frac{1}{e^n},\) error \(\displaystyle <10^{−5}, \sum_{n=1}^∞\frac{1}{e^n}=\frac{1}{e−1}=0.581976…\)

Solution: \(\displaystyle R_N<∫^∞_N\frac{dx}{e^x}=e^{−N},N>5ln(10),\) okay if \(\displaystyle N=12;\sum_{n=1}^{12}e^{−n}=0.581973....\) Estimate agrees with \(\displaystyle 1/(e−1)\) to five decimal places.

43) \(\displaystyle a_n=\frac{1}{e^{n^2}},\) error \(\displaystyle <10^{−5}. \sum_{n=1}^∞n/e^{n^2}=0.40488139857…\)

44) \(\displaystyle a_n=1/n^4,\) error \(\displaystyle <10^{−4}, \sum_{n=1}^∞1/n^4=π^4/90=1.08232...\)

Solution: \(\displaystyle R_N<∫^∞_Ndx/x^4=4/N^3,N>(4.10^4)^{1/3},\) okay if \(\displaystyle N=35\); \(\displaystyle \sum_{n=1}^{35}1/n^4=1.08231….\) Estimate agrees with the sum to four decimal places.

45) \(\displaystyle a_n=1/n^6\), error \(\displaystyle <10^{−6}, \sum_{n=1}^∞1/n^4=π^6/945=1.01734306...,\)

46) Find the limit as \(\displaystyle n→∞\) of \(\displaystyle \frac{1}{n}+\frac{1}{n+1}+⋯+\frac{1}{2n}\). (Hint: Compare to \(\displaystyle ∫^{2n}_n\frac{1}{t}dt.\))

Solution: \(\displaystyle ln(2)\)

47) Find the limit as \(\displaystyle n→∞\) of \(\displaystyle \frac{1}{n}+\frac{1}{n+1}+⋯+\frac{1}{3n}\)

 

The next few exercises are intended to give a sense of applications in which partial sums of the harmonic series arise.

48) In certain applications of probability, such as the so-called Watterson estimator for predicting mutation rates in population genetics, it is important to have an accurate estimate of the number \(\displaystyle H_k=(1+\frac{1}{2}+\frac{1}{3}+⋯+\frac{1}{k})\). Recall that \(\displaystyle T_k=H_k−lnk\) is decreasing. Compute \(\displaystyle T=\lim_{k→∞}T_k\) to four decimal places. (Hint: \(\displaystyle \frac{1}{k+1}<∫^{k+1}_k\frac{1}{x}dx\).)

Solution: \(\displaystyle T=0.5772...\)

49) [T] Complete sampling with replacement, sometimes called the coupon collector’s problem, is phrased as follows: Suppose you have \(\displaystyle N\) unique items in a bin. At each step, an item is chosen at random, identified, and put back in the bin. The problem asks what is the expected number of steps \(\displaystyle E(N)\) that it takes to draw each unique item at least once. It turns out that \(\displaystyle E(N)=N.H_N=N(1+\frac{1}{2}+\frac{1}{3}+⋯+\frac{1}{N})\). Find \(\displaystyle E(N)\) for \(\displaystyle N=10,20,\) and \(\displaystyle 50\).

50) [T] The simplest way to shuffle cards is to take the top card and insert it at a random place in the deck, called top random insertion, and then repeat. We will consider a deck to be randomly shuffled once enough top random insertions have been made that the card originally at the bottom has reached the top and then been randomly inserted. If the deck has \(\displaystyle n\) cards, then the probability that the insertion will be below the card initially at the bottom (call this card \(\displaystyle B\)) is \(\displaystyle 1/n\). Thus the expected number of top random insertions before \(\displaystyle B\) is no longer at the bottom is \(\displaystyle n\). Once one card is below \(\displaystyle B\), there are two places below \(\displaystyle B\) and the probability that a randomly inserted card will fall below \(\displaystyle B\) is \(\displaystyle 2/n\). The expected number of top random insertions before this happens is \(\displaystyle n/2\). The two cards below \(\displaystyle B\) are now in random order. Continuing this way, find a formula for the expected number of top random insertions needed to consider the deck to be randomly shuffled.

Solution: The expected number of random insertions to get \(\displaystyle B\) to the top is \(\displaystyle n+n/2+n/3+⋯+n/(n−1).\) Then one more insertion puts \(\displaystyle B\) back in at random. Thus, the expected number of shuffles to randomize the deck is \(\displaystyle n(1+1/2+⋯+1/n).\)

51) Suppose a scooter can travel \(\displaystyle 100\) km on a full tank of fuel. Assuming that fuel can be transferred from one scooter to another but can only be carried in the tank, present a procedure that will enable one of the scooters to travel \(\displaystyle 100H_N\) km, where \(\displaystyle H_N=1+1/2+⋯+1/N.\)

52) Show that for the remainder estimate to apply on \(\displaystyle [N,∞)\) it is sufficient that \(\displaystyle f(x)\) be decreasing on \(\displaystyle [N,∞)\), but \(\displaystyle f\) need not be decreasing on \(\displaystyle [1,∞).\)

Solution: Set \(\displaystyle b_n=a_{n+N}\) and \(\displaystyle g(t)=f(t+N)\) such that \(\displaystyle f\) is decreasing on \(\displaystyle [t,∞).\)

53) [T] Use the remainder estimate and integration by parts to approximate \(\displaystyle \sum_{n=1}^∞n/e^n\) within an error smaller than \(\displaystyle 0.0001.\)

54) Does \(\displaystyle \sum_{n=2}^∞\frac{1}{n(lnn)^p}\) converge if \(\displaystyle p\) is large enough? If so, for which \(\displaystyle p\)?

Solution: The series converges for \(\displaystyle p>1\) by integral test using change of variable.

55) [T] Suppose a computer can sum one million terms per second of the divergent series \(\displaystyle \sum_{n=1}^N\frac{1}{n}\). Use the integral test to approximate how many seconds it will take to add up enough terms for the partial sum to exceed \(\displaystyle 100\).

56) [T] A fast computer can sum one million terms per second of the divergent series \(\displaystyle \sum_{n=2}^N\frac{1}{nlnn}\). Use the integral test to approximate how many seconds it will take to add up enough terms for the partial sum to exceed \(\displaystyle 100.\)

Solution: \(\displaystyle N=e^{e^{100}}≈e^{10^{43}}\) terms are needed.

9.4: Comparison Tests

Use the comparison test to determine whether the following series converge.

1) \(\displaystyle \sum^∞_{n=1}a_n\) where \(\displaystyle a_n=\frac{2}{n(n+1)}\)

2) \(\displaystyle \sum^∞_{n=1}a_n\) where \(\displaystyle a_n=\frac{1}{n(n+1/2)}\)

Solution: Converges by comparison with \(\displaystyle 1/n^2\).

3) \(\displaystyle \sum^∞_{n=1}\frac{1}{2(n+1)}\)

4) \(\displaystyle \sum^∞_{n=1}\frac{1}{2n−1}\)

Solution: Diverges by comparison with harmonic series, since \(\displaystyle 2n−1≥n.\)

5) \(\displaystyle \sum^∞_{n=2}\frac{1}{(nlnn)^2}\)

6) \(\displaystyle \sum^∞_{n=1}\frac{n!}{(n+2)!}\)

Solution: \(\displaystyle a_n=1/(n+1)(n+2)<1/n^2.\) Converges by comparison with p-series, \(\displaystyle p=2\).

7) \(\displaystyle \sum^∞_{n=1}\frac{1}{n!}\)

8) \(\displaystyle \sum^∞_{n=1}\frac{sin(1/n)}{n}\)

Solution: \(\displaystyle sin(1/n)≤1/n,\) so converges by comparison with p-series, \(\displaystyle p=2\).

9) \(\displaystyle \sum_{n=1}^∞\frac{sin^2n}{n^2}\)

10) \(\displaystyle \sum_{n=1}^∞\frac{sin(1/n)}{\sqrt{n}}\)

Solution: \(\displaystyle sin(1/n)≤1,\) so converges by comparison with p-series, \(\displaystyle p=3/2.\)

11) \(\displaystyle \sum^∞_{n=1}\frac{n^{1.2}−1}{n^{2.3}+1}\)

12) \(\displaystyle \sum^∞_{n=1}\frac{\sqrt{n+1}−\sqrt{n}}{n}\)

Solution: \(\displaystyle Since \sqrt{n+1}−\sqrt{n}=1/(\sqrt{n+1}+\sqrt{n})≤2/\sqrt{n},\) series converges by comparison with p-series for \(\displaystyle p=1.5\).

13) \(\displaystyle \sum^∞_{n=1}\frac{\sqrt[4]{n}}{\sqrt[3]{n^4+n^2}}\)

 

Use the limit comparison test to determine whether each of the following series converges or diverges.

14) \(\displaystyle \sum^∞_{n=1}(\frac{lnn}{n})^2\)

Solution: Converges by limit comparison with p-series for \(\displaystyle p>1\).

15) \(\displaystyle \sum^∞_{n=1}(\frac{lnn}{n^{0.6}})^2\)

16) \(\displaystyle \sum^∞_{n=1}\frac{ln(1+\frac{1}{n})}{n}\)

Solution: Converges by limit comparison with p-series, \(\displaystyle p=2.\)

17) \(\displaystyle \sum^∞_{n=1}ln(1+\frac{1}{n^2})\)

18) \(\displaystyle \sum^∞_{n=1}\frac{1}{4^n−3^n}\)

Solution: Converges by limit comparison with \(\displaystyle 4^{−n}\).

19) \(\displaystyle \sum^∞_{n=1}\frac{1}{n^2−nsinn}\)

20) \(\displaystyle \sum^∞_{n=1}\frac{1}{e^{(1.1)n}−3^n}\)

Solution: Converges by limit comparison with \(\displaystyle 1/e^{1.1n}\).

21) \(\displaystyle \sum^∞_{n=1}\frac{1}{e^{(1.01)n}−3^n}\)

22) \(\displaystyle \sum^∞_{n=1}\frac{1}{n^{1+1/n}}\)

Solution: Diverges by limit comparison with harmonic series.

23) \(\displaystyle \sum^∞_{n=1}\frac{1}{2^{1+1/n}}{n^{1+1/n}}\)

24) \(\displaystyle \sum^∞_{n=1}(\frac{1}{n}−sin(\frac{1}{n}))\)

Solution: Converges by limit comparison with p-series, \(\displaystyle p=3\).

25) \(\displaystyle \sum^∞_{n=1}(1−cos(\frac{1}{n}))\)

26) \(\displaystyle \sum^∞_{n=1}\frac{1}{n}(tan^{−1}n−\frac{π}{2})\)

Solution: Converges by limit comparison with p-series, \(\displaystyle p=3\).

27) \(\displaystyle \sum^∞_{n=1}(1−\frac{1}{n})^{n.n}\) (Hint:\(\displaystyle (1−\frac{1}{n})^n→1/e.\))

28) \(\displaystyle \sum^∞_{n=1}(1−e^{−1/n})\) (Hint:\(\displaystyle 1/e≈(1−1/n)^n,\) so \(\displaystyle 1−e^{−1/n}≈1/n.\))

Solution: Diverges by limit comparison with \(\displaystyle 1/n\).

29) Does \(\displaystyle \sum^∞_{n=2}\frac{1}{(lnn)^p}\) converge if \(\displaystyle p\) is large enough? If so, for which \(\displaystyle p?\)

30) Does \(\displaystyle \sum^∞_{n=1}(\frac{(lnn)}{n})^p\) converge if \(\displaystyle p\) is large enough? If so, for which \(\displaystyle p?\)

Solution: Converges for \(\displaystyle p>1\) by comparison with a \(\displaystyle p\) series for slightly smaller \(\displaystyle p\).

31) For which \(\displaystyle p\) does the series \(\displaystyle \sum^∞_{n=1}2^{pn}/3^n\) converge?

32) For which \(\displaystyle p>0\) does the series \(\displaystyle \sum^∞_{n=1}\frac{n^p}{2^n}\) converge?

Solution: Converges for all \(\displaystyle p>0\).

33) For which \(\displaystyle r>0\) does the series \(\displaystyle \sum^∞_{n=1}\frac{r^{n^2}}{2^n}\) converge?

34) For which \(\displaystyle r>0\) does the series \(\displaystyle \sum^∞_{n=1}\frac{2^n}{r^{n^2}}\) converge?

Solution: Converges for all \(\displaystyle r>1\). If \(\displaystyle r>1\) then \(\displaystyle r^n>4\), say, once \(\displaystyle n>ln(2)/ln(r)\) and then the series converges by limit comparison with a geometric series with ratio \(\displaystyle 1/2\).

35) Find all values of \(\displaystyle p\) and \(\displaystyle q\) such that \(\displaystyle \sum^∞_{n=1}\frac{n^p}{(n!)^q}\) converges.

36) Does \(\displaystyle \sum^∞_{n=1}\frac{sin^2(nr/2)}{n}\) converge or diverge? Explain.

Solution: The numerator is equal to \(\displaystyle 1\) when \(\displaystyle n\) is odd and \(\displaystyle 0\) when \(\displaystyle n\) is even, so the series can be rewritten \(\displaystyle \sum^∞_{n=1}\frac{1}{2n+1},\) which diverges by limit comparison with the harmonic series.

37) Explain why, for each \(\displaystyle n\), at least one of \(\displaystyle {|sinn|,|sin(n+1)|,...,|sinn+6|}\) is larger than \(\displaystyle 1/2\). Use this relation to test convergence of \(\displaystyle \sum^∞_{n=1}\frac{|sinn|}{\sqrt{n}}\).

38) Suppose that \(\displaystyle a_n≥0\) and \(\displaystyle b_n≥0\) and that \(\displaystyle \sum_{n=1}^∞a^2_n\) and \(\displaystyle \sum_{n=1}^∞b^2_n\) converge. Prove that \(\displaystyle \sum_{n=1}^∞a_nb_n\) converges and \(\displaystyle \sum_{n=1}^∞a_nb_n≤\frac{1}{2}(\sum_{n=1}^∞a^2_n+\sum_{n=1}^∞b^2_n)\).

Solution: \(\displaystyle (a−b)^2=a^2−2ab+b^2\) or \(\displaystyle a^2+b^2≥2ab\), so convergence follows from comparison of \(\displaystyle 2a_nb_n\) with \(\displaystyle a^2_n+b^2_n.\) Since the partial sums on the left are bounded by those on the right, the inequality holds for the infinite series.

39) Does \(\displaystyle \sum_{n=1}^∞2^{−lnlnn}\) converge? (Hint: Write \(\displaystyle 2^{lnlnn}\) as a power of \(\displaystyle lnn\).)

40) Does \(\displaystyle \sum_{n=1}^∞(lnn)^{−lnn}\) converge? (Hint: Use \(\displaystyle t=e^{ln(t)}\) to compare to a \(\displaystyle p−series\).)

Solution: \(\displaystyle (lnn)^{−lnn}=e^{−ln(n)lnln(n)}.\) If \(\displaystyle n\) is sufficiently large, then \(\displaystyle lnlnn>2,\) so \(\displaystyle (lnn)^{−lnn}<1/n^2\), and the series converges by comparison to a \(\displaystyle p−series.\)

41) Does \(\displaystyle \sum_{n=2}^∞(lnn)^{−lnlnn}\) converge? (Hint: Compare \(\displaystyle a_n\) to \(\displaystyle 1/n\).)

42) Show that if \(\displaystyle a_n≥0\) and \(\displaystyle \sum_{n=1}^∞a_n\) converges, then \(\displaystyle \sum_{n=1}^∞a^2_n\) converges. If \(\displaystyle \sum_{n=1}^∞a^2_n\) converges, does \(\displaystyle \sum_{n=1}^∞a_n\) necessarily converge?

Solution: \(\displaystyle a_n→0,\) so \(\displaystyle a^2_n≤|a_n|\) for large \(\displaystyle n\). Convergence follows from limit comparison. \(\displaystyle \sum1/n^2\) converges, but \(\displaystyle \sum1/n\) does not, so the fact that \(\displaystyle \sum_{n=1}^∞a^2_n\) converges does not imply that \(\displaystyle \sum_{n=1}^∞a_n\) converges.

43) Suppose that \(\displaystyle a_n>0\) for all \(\displaystyle n\) and that \(\displaystyle \sum_{n=1}^∞a_n\) converges. Suppose that \(\displaystyle b_n\) is an arbitrary sequence of zeros and ones. Does \(\displaystyle \sum_{n=1}^∞a_nb_n\) necessarily converge?

44) Suppose that \(\displaystyle a_n>0\) for all \(\displaystyle n\) and that \(\displaystyle \sum_{n=1}^∞a_n\) diverges. Suppose that \(\displaystyle b_n\) is an arbitrary sequence of zeros and ones with infinitely many terms equal to one. Does \(\displaystyle \sum_{n=1}^∞a_nb_n\) necessarily diverge?

Solution: No. \(\displaystyle \sum_{n=1}^∞1/n\) diverges. Let \(\displaystyle b_k=0\) unless \(\displaystyle k=n^2\) for some \(\displaystyle n\). Then \(\displaystyle \sum_kb_k/k=\sum1/k^2\) converges.

45) Complete the details of the following argument: If \(\displaystyle \sum_{n=1}^∞\frac{1}{n}\) converges to a finite sum \(\displaystyle s\), then \(\displaystyle \frac{1}{2}s=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+⋯\) and \(\displaystyle s−\frac{1}{2}s=1+\frac{1}{3}+\frac{1}{5}+⋯.\) Why does this lead to a contradiction?

46) Show that if \(\displaystyle a_n≥0\) and \(\displaystyle \sum_{n=1}^∞a^2_n\) converges, then \(\displaystyle \sum_{n=1}^∞sin^2(a_n)\) converges.

Solution: \(\displaystyle |sint|≤|t|,\) so the result follows from the comparison test.

47) Suppose that \(\displaystyle a_n/b_n→0\) in the comparison test, where \(\displaystyle a_n≥0\) and \(\displaystyle b_n≥0\). Prove that if \(\displaystyle \sum b_n\) converges, then \(\displaystyle \sum a_n\) converges.

48) Let \(\displaystyle b_n\) be an infinite sequence of zeros and ones. What is the largest possible value of \(\displaystyle x=\sum_{n=1}^∞b_n/2^n\)?

Solution: By the comparison test, \(\displaystyle x=\sum_{n=1}^∞b_n/2^n≤\sum_{n=1}^∞1/2^n=1.\)

49) Let \(\displaystyle d_n\) be an infinite sequence of digits, meaning \(\displaystyle d_n\) takes values in \(\displaystyle {0,1,…,9}\). What is the largest possible value of \(\displaystyle x=\sum_{n=1}^∞d_n/10^n\) that converges?

50) Explain why, if \(\displaystyle x>1/2,\) then \(\displaystyle x\) cannot be written \(\displaystyle x=\sum_{n=2}^∞\frac{b_n}{2^n}(b_n=0or1,b_1=0).\)

Solution: If \(\displaystyle b_1=0,\) then, by comparison, \(\displaystyle x≤\sum_{n=2}^∞1/2^n=1/2.\)

51) [T] Evelyn has a perfect balancing scale, an unlimited number of \(\displaystyle 1-kg\) weights, and one each of \(\displaystyle 1/2-kg,1/4-kg,1/8-kg,\) and so on weights. She wishes to weigh a meteorite of unspecified origin to arbitrary precision. Assuming the scale is big enough, can she do it? What does this have to do with infinite series?

52) [T] Robert wants to know his body mass to arbitrary precision. He has a big balancing scale that works perfectly, an unlimited collection of \(\displaystyle 1-kg\) weights, and nine each of \(\displaystyle 0.1-kg, 0.01-kg,0.001-kg,\) and so on weights. Assuming the scale is big enough, can he do this? What does this have to do with infinite series?

Solution: Yes. Keep adding \(\displaystyle 1-kg\) weights until the balance tips to the side with the weights. If it balances perfectly, with Robert standing on the other side, stop. Otherwise, remove one of the \(\displaystyle 1-kg\) weights, and add \(\displaystyle 0.1-kg\) weights one at a time. If it balances after adding some of these, stop. Otherwise if it tips to the weights, remove the last \(\displaystyle 0.1-kg\) weight. Start adding \(\displaystyle 0.01-kg\) weights. If it balances, stop. If it tips to the side with the weights, remove the last \(\displaystyle 0.01-kg\) weight that was added. Continue in this way for the \(\displaystyle 0.001-kg\) weights, and so on. After a finite number of steps, one has a finite series of the form \(\displaystyle A+\sum_{n=1}^Ns_n/10^n\) where \(\displaystyle A\) is the number of full kg weights and \(\displaystyle d_n\) is the number of \(\displaystyle 1/10^n-kg\) weights that were added. If at some state this series is Robert’s exact weight, the process will stop. Otherwise it represents the Nth partial sum of an infinite series that gives Robert’s exact weight, and the error of this sum is at most \(\displaystyle 1/10^N\).

53) The series \(\displaystyle \sum_{n=1}^∞\frac{1}{2n}\) is half the harmonic series and hence diverges. It is obtained from the harmonic series by deleting all terms in which \(\displaystyle n\) is odd. Let \(\displaystyle m>1\) be fixed. Show, more generally, that deleting all terms \(\displaystyle 1/n\) where \(\displaystyle n=mk\) for some integer \(\displaystyle k\) also results in a divergent series.

54) In view of the previous exercise, it may be surprising that a subseries of the harmonic series in which about one in every five terms is deleted might converge. A depleted harmonic series is a series obtained from \(\displaystyle \sum_{n=1}^∞\frac{1}{n}\) by removing any term \(\displaystyle 1/n\) if a given digit, say \(\displaystyle 9\), appears in the decimal expansion of \(\displaystyle n\).Argue that this depleted harmonic series converges by answering the following questions.

a. How many whole numbers \(\displaystyle n\) have \(\displaystyle d\) digits?

b. How many \(\displaystyle d-digit\) whole numbers \(\displaystyle h(d)\). do not contain \(\displaystyle 9\) as one or more of their digits?

c. What is the smallest \(\displaystyle d-digit\) number \(\displaystyle m(d)\)?

d. Explain why the deleted harmonic series is bounded by \(\displaystyle \sum_{d=1}^∞\frac{h(d)}{m(d)}\).

e. Show that \(\displaystyle \sum_{d=1}^∞\frac{h(d)}{m(d)}\) converges.

Solution: a. \(\displaystyle 10^d−10^{d−1}<10^d\) b. \(\displaystyle h(d)<9^d\) c. \(\displaystyle m(d)=10^{d−1}+1\) d. Group the terms in the deleted harmonic series together by number of digits. \(\displaystyle h(d)\) bounds the number of terms, and each term is at most \(\displaystyle 1/m(d). \sum_{d=1}^∞h(d)/m(d)≤\sum_{d=1}^∞9^d/(10)^{d−1}≤90\). One can actually use comparison to estimate the value to smaller than \(\displaystyle 80\). The actual value is smaller than \(\displaystyle 23\).

55) Suppose that a sequence of numbers \(\displaystyle a_n>0\) has the property that \(\displaystyle a_1=1\) and \(\displaystyle a_{n+1}=\frac{1}{n+1}S_n\), where \(\displaystyle S_n=a_1+⋯+a_n\). Can you determine whether \(\displaystyle \sum_{n=1}^∞a_n\) converges? (Hint: \(\displaystyle S_n\) is monotone.)

56) Suppose that a sequence of numbers \(\displaystyle a_n>0\) has the property that \(\displaystyle a_1=1\) and \(\displaystyle a_{n+1}=\frac{1}{(n+1)^2}S_n\), where \(\displaystyle S_n=a_1+⋯+a_n\). Can you determine whether \(\displaystyle \sum_{n=1}^∞a_n\) converges? (Hint: \(\displaystyle S_2=a_2+a_1=a_2+S_1=a_2+1=1+1/4=(1+1/4)S_1, S_3=\frac{1}{3^2}S_2+S_2=(1+1/9)S_2=(1+1/9)(1+1/4)S_1\), etc. Look at \(\displaystyle ln(S_n)\), and use \(\displaystyle ln(1+t)≤t, t>0.\))

Solution: Continuing the hint gives \(\displaystyle S_N=(1+1/N^2)(1+1/(N−1)^2…(1+1/4)).\) Then \(\displaystyle ln(S_N)=ln(1+1/N^2)+ln(1+1/(N−1)^2)+⋯+ln(1+1/4).\) Since \(\displaystyle ln(1+t)\) is bounded by a constant times \(\displaystyle t\), when \(\displaystyle 0<t<1\) one has \(\displaystyle ln(S_N)≤C\sum_{n=1}^N\frac{1}{n^2}\), which converges by comparison to the p-series for \(\displaystyle p=2\).

9.5: Alternating Series

State whether each of the following series converges absolutely, conditionally, or not at all.

1) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}\frac{n}{n+3}\)

2) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}\frac{\sqrt{n}+1}{\sqrt{n}+3}\)

Solution: Does not converge by divergence test. Terms do not tend to zero.

3) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}\frac{1}{\sqrt{n+3}}\)

4) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}\frac{\sqrt{n+3}}{n}\)

Solution: Converges conditionally by alternating series test, since \(\displaystyle \sqrt{n+3}/n\) is decreasing. Does not converge absolutely by comparison with p-series, \(\displaystyle p=1/2\).

5) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}\frac{1}{n!}\)

6) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}\frac{3^n}{n!}\)

Solution: Converges absolutely by limit comparison to \(\displaystyle 3^n/4^n,\) for example.

7) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}(\frac{n−1}{n})^n\)

8) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}(\frac{n+1}{n})^n\)

Solution: Diverges by divergence test since \(\displaystyle \lim_{n→∞}|a_n|=e\).

9) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}sin^2n\)

10) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}cos^2n\)

Solution: Does not converge. Terms do not tend to zero.

11) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}sin^2(1/n)\)

12) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}cos^2(1/n)\)

Solution: \(\displaystyle \lim_{n→∞}cos^2(1/n)=1.\) Diverges by divergence test.

13) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}ln(1/n)\)

14) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}ln(1+\frac{1}{n})\)

Solution: Converges by alternating series test.

15) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}\frac{n^2}{1+n^4}\)

16) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}\frac{n^e}{1+n^π}\)

Solution: Converges conditionally by alternating series test. Does not converge absolutely by limit comparison with p-series, \(\displaystyle p=π−e\)

17) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}2^{1/n}\)

18) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}n^{1/n}\)

Solution: Diverges; terms do not tend to zero.

19) \(\displaystyle \sum^∞_{n=1}(−1)^n(1−n^{1/n})\) (Hint: \(\displaystyle n^{1/n}≈1+ln(n)/n\) for large \(\displaystyle n\).)

20) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}n(1−cos(\frac{1}{n}))\) (Hint: \(\displaystyle cos(1/n)≈1−1/n^2\) for large \(\displaystyle n\).)

Solution: Converges by alternating series test. Does not converge absolutely by limit comparison with harmonic series.

21) \(\displaystyle \sum^∞_{n=1}(−1)^n+1(\sqrt{n+1}−\sqrt{n})\) (Hint: Rationalize the numerator.)

22) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}(\frac{1}{\sqrt{n}}−\frac{1}{\sqrt{n+1}})\) (Hint: Cross-multiply then rationalize numerator.)

Solution: Converges absolutely by limit comparison with p-series, \(\displaystyle p=3/2\), after applying the hint.

23) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}(ln(n+1)−lnn)\)

24) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}n(tan^{−1}(n+1)−tan^{−1}n)\) (Hint: Use Mean Value Theorem.)

Solution: Converges by alternating series test since \(\displaystyle n(tan^{−1}(n+1)−tan^{−1}n)\) is decreasing to zero for large \(\displaystyle n\).Does not converge absolutely by limit comparison with harmonic series after applying hint.

25) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}((n+1)^2−n^2)\)

26) \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}(\frac{1}{n}−\frac{1}{n+1})\)

Solution: Converges absolutely, since \(\displaystyle a_n=\frac{1}{n}−\frac{1}{n+1}\) are terms of a telescoping series.

27) \(\displaystyle \sum^∞_{n=1}\frac{cos(nπ)}{n}\)

28) \(\displaystyle \sum^∞_{n=1}\frac{cos(nπ)}{n^{1/n}}\)

Solution: Terms do not tend to zero. Series diverges by divergence test.

29) \(\displaystyle \sum^∞_{n=1}\frac{1}{n}sin(\frac{nπ}{2})\)

30) \(\displaystyle \sum^∞_{n=1}sin(nπ/2)sin(1/n)\)

Solution: Converges by alternating series test. Does not converge absolutely by limit comparison with harmonic series.

 

In each of the following problems, use the estimate \(\displaystyle |R_N|≤b_{N+1}\) to find a value of \(\displaystyle N\) that guarantees that the sum of the first N terms of the alternating series \(\displaystyle \sum^∞_{n=1}(−1)^{n+1}b_n\) differs from the infinite sum by at most the given error. Calculate the partial sum \(\displaystyle S_N\) for this \(\displaystyle N\).

31) [T] \(\displaystyle b_n=1/n,\) error \(\displaystyle <10^{−5}\)

32) [T] \(\displaystyle b_n=1/ln(n), n≥2,\) error \(\displaystyle <10^{−1}\)

Solution: \(\displaystyle ln(N+1)>10, N+1>e^{10}, N≥22026; S_{22026}=0.0257…\)

33) [T] \(\displaystyle b_n=1/\sqrt{n},\) error \(\displaystyle <10^{−3}\)

34) [T] \(\displaystyle b_n=1/2^n\), error \(\displaystyle <10^{−6}\)

Solution: \(\displaystyle 2^{N+1}>10^6\) or \(\displaystyle N+1>6ln(10)/ln(2)=19.93.\) or \(\displaystyle N≥19; S_{19}=0.333333969…\)

35) [T] \(\displaystyle b_n=ln(1+\frac{1}{n}),\) error \(\displaystyle <10^{−3}\)

36) [T] \(\displaystyle b_n=1/n^2,\) error \(\displaystyle <10^{−6}\)

Solution: \(\displaystyle (N+1)^2>10^6\) or \(\displaystyle N>999; S_{1000}≈0.822466.\)

 

For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false.

37) If \(\displaystyle b_n≥0\) is decreasing and \(\displaystyle \lim_{n→∞}b_n=0\), then \(\displaystyle \sum_{n=1}^∞(b_{2n−1}−b_{2n})\) converges absolutely.

38) If \(\displaystyle b_n≥0\) is decreasing, then \(\displaystyle \sum_{n=1}^∞(b_{2n−1}−b_{2n})\) converges absolutely.

Solution: True. \(\displaystyle b_n\) need not tend to zero since if \(\displaystyle c_n=b_n−\lim b_n\), then \(\displaystyle c_{2n−1}−c_{2n}=b_{2n−1}−b_{2n}.\)

39) If \(\displaystyle b_n≥0\) and \(\displaystyle \lim_{n→∞}b_n=0\) then \(\displaystyle \sum_{n=1}^∞(\frac{1}{2}(b_{3n−2}+b_{3n−1})−b_{3n})\) converges.

40) If \(\displaystyle b_n≥0\) is decreasing and \(\displaystyle \sum_{n=1}^∞(b_{3n−2}+b_{3n−1}−b_{3n})\) converges then \(\displaystyle \sum_{n=1}^∞b_{3n−2}\) converges.

Solution: True. \(\displaystyle b_{3n−1}−b_{3n}≥0,\) so convergence of \(\displaystyle \sum b_{3n−2}\) follows from the comparison test.

41) If \(\displaystyle b_n≥0\) is decreasing and \(\displaystyle \sum_{n=1}^∞(−1)^{n−1}b_n\) converges conditionally but not absolutely, then \(\displaystyle b_n\) does not tend to zero.

42) Let \(\displaystyle a^+_n=a_n\) if \(\displaystyle a_n≥0\) and \(\displaystyle a^−_n=−a_n\) if \(\displaystyle a_n<0\). (Also, \(\displaystyle a+n=0\) if \(\displaystyle a_n<0\) and \(\displaystyle a−n=0\) if \(\displaystyle a_n≥0\).) If \(\displaystyle \sum_{n=1}^∞a_n\) converges conditionally but not absolutely, then neither \(\displaystyle \sum_{n=1}^∞a^+_n\) nor \(\displaystyle \sum_{n=1}^∞a^−_n\) converge.

Solution: True. If one converges, then so must the other, implying absolute convergence.

43) Suppose that \(\displaystyle a_n\) is a sequence of positive real numbers and that \(\displaystyle \sum_{n=1}^∞a_n\) converges.

44) Suppose that \(\displaystyle b_n\) is an arbitrary sequence of ones and minus ones. Does \(\displaystyle \sum_{n=1}^∞a_nb_n\) necessarily converge?

45) Suppose that \(\displaystyle a_n\) is a sequence such that \(\displaystyle \sum_{n=1}^∞a_nb_n\) converges for every possible sequence \(\displaystyle b_n\) of zeros and ones. Does \(\displaystyle \sum_{n=1}^∞a_n\) converge absolutely?

Solution: Yes. Take \(\displaystyle b_n=1\) if \(\displaystyle a_n≥0\) and \(\displaystyle b_n=0\) if \(\displaystyle a_n<0\). Then \(\displaystyle \sum_{n=1}^∞a_nb_n=\sum_{n:an≥0}a_n\) converges. Similarly, one can show \(\displaystyle \sum_{n:an<0}a_n\) converges. Since both series converge, the series must converge absolutely.

 

The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.

46) \(\displaystyle \sum_{n=1}^∞(−1)^{n+1}\frac{sin^2n}{n}\)

47) \(\displaystyle \sum_{n=1}^∞(−1)^{n+1}\frac{cos^2n}{n}\)

Solution: Not decreasing. Does not converge absolutely.

48) \(\displaystyle 1+\frac{1}{2}−\frac{1}{3}−\frac{1}{4}+\frac{1}{5}+\frac{1}{6}−\frac{1}{7}−\frac{1}{8}+⋯\)

49) \(\displaystyle 1+\frac{1}{2}−\frac{1}{3}+\frac{1}{4}+\frac{1}{5}−\frac{1}{6}+\frac{1}{7}+\frac{1}{8}−\frac{1}{9}+⋯\)

Solution: Not alternating. Can be expressed as \(\displaystyle \sum_{n=1}^∞(\frac{1}{3n−2}+\frac{1}{3n−1}−\frac{1}{3n}),\) which diverges by comparison with \(\displaystyle \sum\frac{1}{3n−2}.\)

50) Show that the alternating series \(\displaystyle 1−\frac{1}{2}+\frac{1}{2}−\frac{1}{4}+\frac{1}{3}−\frac{1}{6}+\frac{1}{4}−\frac{1}[8]+⋯\) does not converge. What hypothesis of the alternating series test is not met?

51) Suppose that \(\displaystyle \sum a_n\) converges absolutely. Show that the series consisting of the positive terms \(\displaystyle a_n\) also converges.

Solution: Let \(\displaystyle a^+_n=a_n\) if \(\displaystyle a_n≥0\) and \(\displaystyle a^+_n=0\) if \(\displaystyle a_n<0\). Then \(\displaystyle a^+_n≤|a_n|\) for all \(\displaystyle n\) so the sequence of partial sums of \(\displaystyle a^+_n\) is increasing and bounded above by the sequence of partial sums of \(\displaystyle |a_n|\), which converges; hence, \(\displaystyle \sum_{n=1}^∞a^+_n\) converges.

52) Show that the alternating series \(\displaystyle \frac{2}{3}−\frac{3}{5}+\frac{4}{7}−\frac{5}{9}+⋯\) does not converge. What hypothesis of the alternating series test is not met?

53) The formula \(\displaystyle cosθ=1−\frac{θ^2}{2!}+\frac{θ^4}{4!}−\frac{θ^6}{6!}+⋯\) will be derived in the next chapter. Use the remainder \(\displaystyle |R_N|≤b_{N+1}\) to find a bound for the error in estimating \(\displaystyle cosθ\) by the fifth partial sum \(\displaystyle 1−θ^2/2!+θ^4/4!−θ^6/6!+θ^8/8!\) for \(\displaystyle θ=1, θ=π/6,\) and \(\displaystyle θ=π.\)

Solution: For \(\displaystyle N=5\) one has \(\displaystyle ∣R_N∣b_6=θ^{10}/10!\). When \(\displaystyle θ=1, R_5≤1/10!≈2.75×10^{−7}\). When \(\displaystyle θ=π/6,\) \(\displaystyle R_5≤(π/6)^{10}/10!≈4.26×10^{−10}\). When \(\displaystyle θ=π, R_5≤π^{10}/10!=0.0258.\)

54) The formula \(\displaystyle sinθ=θ−\frac{θ^3}{3!}+\frac{θ^5}{5!}−\frac{θ^7}{7!}+⋯\) will be derived in the next chapter. Use the remainder \(\displaystyle |R_N|≤b_{N+1}\) to find a bound for the error in estimating \(\displaystyle sinθ\) by the fifth partial sum \(\displaystyle θ−θ^3/3!+θ^5/5!−θ^7/7!+θ^9/9!\) for \(\displaystyle θ=1, θ=π/6,\) and \(\displaystyle θ=π.\)

55) How many terms in \(\displaystyle cosθ=1−\frac{θ^2}{2!}+\frac{θ^4}{4!}−\frac{θ^6}{6!}+⋯\) are needed to approximate \(\displaystyle cos1\) accurate to an error of at most \(\displaystyle 0.00001\)?

Solution: Let \(\displaystyle b_n=1/(2n−2)!.\) Then \(\displaystyle R_N≤1/(2N)!<0.00001\) when \(\displaystyle (2N)!>10^5\) or \(\displaystyle N=5\) and \(\displaystyle 1−\frac{1}{2!}+\frac{1}{4!}−\frac{1}{6!}+\frac{1}{8!}=0.540325…\), whereas \(\displaystyle cos1=0.5403023…\)

56) How many terms in \(\displaystyle sinθ=θ−\frac{θ^3}{3!}+\frac{θ^5}{5!}−\frac{θ^7}{7!}+⋯\) are needed to approximate \(\displaystyle sin1\) accurate to an error of at most \(\displaystyle 0.00001?\)

57) Sometimes the alternating series \(\displaystyle \sum_{n=1}^∞(−1)^{n−1}b_n\) converges to a certain fraction of an absolutely convergent series \(\displaystyle \sum_{n=1}^∞b_n\) at a faster rate. Given that \(\displaystyle \sum_{n=1}^∞\frac{1}{n^2}=\frac{π^2}{6}\), find \(\displaystyle S=1−\frac{1}{2^2}+\frac{1}{3^2}−\frac{1}{4^2}+⋯\). Which of the series \(\displaystyle 6\sum_{n=1}^∞\frac{1}{n^2}\) and \(\displaystyle S\sum_{n=1}^∞\frac{(−1)^{n−1}}{n^2}\) gives a better estimation of \(\displaystyle π^2\) using \(\displaystyle 1000\) terms?

Solution: Let \(\displaystyle T=\sum\frac{1}{n^2}.\) Then \(\displaystyle T−S=\frac{1}{2}T\), so \(\displaystyle S=T/2\). \(\displaystyle \sqrt{6×\sum_{n=1}^{1000}1/n^2}=3.140638…; \sqrt{\frac{1}{2}×\sum_{n=1}^{1000}(−1)^{n−1}/n^2}=3.141591…; π=3.141592….\) The alternating series is more accurate for \(\displaystyle 1000\) terms.

 

The following alternating series converge to given multiples of \(\displaystyle π\). Find the value of \(\displaystyle N\) predicted by the remainder estimate such that the \(\displaystyle Nth\) partial sum of the series accurately approximates the left-hand side to within the given error. Find the minimum \(\displaystyle N\) for which the error bound holds, and give the desired approximate value in each case. Up to \(\displaystyle 15\) decimals places, \(\displaystyle π=3.141592653589793….\)

58) [T] \(\displaystyle \frac{π}{4}=\sum_{n=0}^∞\frac{(−1)^n}{2n+1},\) error \(\displaystyle <0.0001\)

59) [T] \(\displaystyle \frac{π}{\sqrt{12}}=\sum_{k=0}^∞\frac{(−3)^{−k}}{2k+1},\) error \(\displaystyle <0.0001\)

Solution: \(\displaystyle N=6, S_N=0.9068\)

60) [T] The series \(\displaystyle \sum_{n=0}^∞\frac{sin(x+πn)}{x+πn}\) plays an important role in signal processing. Show that \(\displaystyle \sum_{n=0}^∞\frac{sin(x+πn)}{x+πn}\) converges whenever \(\displaystyle 0<x<π\). (Hint: Use the formula for the sine of a sum of angles.)

61) [T] If \(\displaystyle \sum_{n=1}^N(−1)^{n−1}\frac{1}{n}→ln2,\) what is \(\displaystyle 1+\frac{1}{3}+\frac{1}{5}−\frac{1}{2}−\frac{1}{4}−\frac{1}{6}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}−\frac{1}{8}−\frac{1}{10}−\frac{1}{12}+⋯?\)

Solution: \(\displaystyle ln(2).\) The \(\displaystyle 3nth\) partial sum is the same as that for the alternating harmonic series.

62) [T] Plot the series \(\displaystyle \sum_{n=1}^{100}\frac{cos(2πnx)}{n}\) for \(\displaystyle 0≤x<1\). Explain why \(\displaystyle \sum_{n=1}^{100}\frac{cos(2πnx)}{n}\) diverges when \(\displaystyle x=0,1\). How does the series behave for other \(\displaystyle x\)?

63) [T] Plot the series \(\displaystyle \sum_{n=1}^{100}\frac{sin(2πnx)}{n}\) for \(\displaystyle 0≤x<1\) and comment on its behavior

Solution: The series jumps rapidly near the endpoints. For \(\displaystyle x\) away from the endpoints, the graph looks like \(\displaystyle π(1/2−x)\).

64) [T] Plot the series \(\displaystyle \sum_{n=1}^{100}\frac{cos(2πnx)}{n^2}\) for \(\displaystyle 0≤x<1\) and describe its graph.

65) [T] The alternating harmonic series converges because of cancellation among its terms. Its sum is known because the cancellation can be described explicitly. A random harmonic series is one of the form \(\displaystyle \sum_{n=1}^∞\frac{S_n}{n}\), where \(\displaystyle s_n\) is a randomly generated sequence of \(\displaystyle ±1's\) in which the values \(\displaystyle ±1\) are equally likely to occur. Use a random number generator to produce \(\displaystyle 1000\) random \(\displaystyle ±1s\) and plot the partial sums \(\displaystyle S_N=\sum_{n=1}^N\frac{s_n}{n}\) of your random harmonic sequence for \(\displaystyle N=1\) to \(\displaystyle 1000\). Compare to a plot of the first \(\displaystyle 1000\) partial sums of the harmonic series.

Solution: Here is a typical result. The top curve consists of partial sums of the harmonic series. The bottom curve plots partial sums of a random harmonic series.

66) [T] Estimates of \(\displaystyle \sum_{n=1}^∞\frac{1}{n^2}\) can be accelerated by writing its partial sums as \(\displaystyle \sum_{n=1}^N\frac{1}{n^2}=\sum_{n=1}^N\frac{1}{n(n+1)}+\sum_{n=1}^N\frac{1}{n^2(n+1)}\) and recalling that \(\displaystyle \sum_{n=1}^N\frac{1}{n(n+1)}=1−\frac{1}{N+1}\) converges to one as \(\displaystyle N→∞.\) Compare the estimate of \(\displaystyle π^2/6\) using the sums \(\displaystyle \sum_{n=1}^{1000}\frac{1}{n^2}\) with the estimate using \(\displaystyle 1+\sum_{n=1}^{1000}\frac{1}{n^2(n+1)}\).

67) [T] The Euler transform rewrites \(\displaystyle S=\sum_{n=0}^∞(−1)^nb_n\) as \(\displaystyle S=\sum_{n=0}^∞(−1)^n2^{−n−1}\sum_{m=0}^n(^n_m)b_{n−m}\). For the alternating harmonic series, it takes the form \(\displaystyle ln(2)=\sum_{n=1}^∞\frac{(−1)^{n−1}}{n}=\sum_{n=1}^∞\frac{1}{n2^n}\). Compute partial sums of \(\displaystyle \sum_{n=1}^∞\frac{1}{n2^n}\) until they approximate \(\displaystyle ln(2)\) accurate to within \(\displaystyle 0.0001\). How many terms are needed? Compare this answer to the number of terms of the alternating harmonic series are needed to estimate \(\displaystyle ln(2)\).

Solution: By the alternating series test, \(\displaystyle |S_n−S|≤b_{n+1},\) so one needs \(\displaystyle 10^4\) terms of the alternating harmonic series to estimate \(\displaystyle ln(2)\) to within \(\displaystyle 0.0001\). The first \(\displaystyle 10\) partial sums of the series \(\displaystyle \sum_{n=1}^∞\frac{1}{n2^n}\) are (up to four decimals) \(\displaystyle 0.5000,0.6250,0.6667,0.6823,0.6885,0.6911,0.6923,0.6928,0.6930,0.6931\) and the tenth partial sum is within \(\displaystyle 0.0001\) of \(\displaystyle ln(2)=0.6931….\)

68) [T] In the text it was stated that a conditionally convergent series can be rearranged to converge to any number. Here is a slightly simpler, but similar, fact. If \(\displaystyle a_n≥0\) is such that \(\displaystyle a_n→0\) as \(\displaystyle n→∞\) but \(\displaystyle \sum_{n=1}^∞a_n\) diverges, then, given any number \(\displaystyle A\) there is a sequence \(\displaystyle s_n\) of \(\displaystyle ±1's\) such that \(\displaystyle \sum_{n=1}^∞a_ns_n→A.\) Show this for \(\displaystyle A>0\) as follows.

a. Recursively define \(\displaystyle s_n\) by \(\displaystyle s_n=1\) if \(\displaystyle S_{n−1}=\sum_{k=1}^{n−1}a_ks_k<A\) and \(\displaystyle s_n=−1\) otherwise.

b. Explain why eventually \(\displaystyle S_n≥A,\) and for any \(\displaystyle m\) larger than this \(\displaystyle n\), \(\displaystyle A−a_m≤S_m≤A+a_m\).

c. Explain why this implies that \(\displaystyle S_n→A\) as \(\displaystyle n→∞.\)

9.6: Ratio and Root Tests

Use the ratio test to determine whether \(\displaystyle \sum^∞_{n=1}a_n\) converges, where \(\displaystyle a_n\) is given in the following problems. State if the ratio test is inconclusive.

1) \(\displaystyle a_n=1/n!\)

Solution: \(\displaystyle a_{n+1}/a_n→0.\) Converges.

2) \(\displaystyle a_n=10^n/n!\)

3) \(\displaystyle a_n=n^2/2^n\)

Solution: \(\displaystyle \frac{a_{n+1}}{a_n}=\frac{1}{2}(\frac{n+1}{n})^2→1/2<1.\) Converges.

4) \(\displaystyle a_n=n^{10}/2^n\)

5) \(\displaystyle \sum_{n=1}^∞\frac{(n!)^3}{(3n!)}\)

Solution: \(\displaystyle \frac{a_{n+1}}{a_n}→1/27<1.\) Converges.

6) \(\displaystyle \sum_{n=1}^∞\frac{2^{3n}(n!)^3}{(3n!)}\)

7) \(\displaystyle \sum_{n=1}^∞\frac{(2n)!}{n^{2n}}\)

Solution: \(\displaystyle \frac{a_{n+1}}{a_n}→4/e^2<1.\) Converges.

8) \(\displaystyle \sum_{n=1}^∞\frac{(2n)!}{(2n)^n}\)

9) \(\displaystyle \sum_{n=1}^∞\frac{n!}{(n/e)^n}\)

Solution: \(\displaystyle \frac{a_{n+1}}{a_n}→1.\) Ratio test is inconclusive.

10) \(\displaystyle \sum_{n=1}^∞\frac{(2n)!}{(n/e)^{2n}}\)

11) \(\displaystyle \sum_{n=1}^∞\frac{(2^nn!)^2}{(2n)^{2n}}\)

Solution: \(\displaystyle \frac{a_n}{a_{n+1}}→1/e^2.\) Converges.

 

Use the root test to determine whether \(\displaystyle \sum^∞_{n=1}a_n\) converges, where \(\displaystyle a_n\) is as follows.

12) \(\displaystyle a_k=(\frac{k−1}{2k+3})^k\)

13) \(\displaystyle a_k=\frac{(2k^2−1}{k^2+3})^k\)

Solution: \(\displaystyle (a_k)^{1/k}→2>1.\) Diverges.

14) \(\displaystyle a_n=\frac{(lnn)^{2n}}{n^n}\)

15) \(\displaystyle a_n=n/2^n\)

Solution: \(\displaystyle (a_n)^{1/n}→1/2<1.\) Converges.

16) \(\displaystyle a_n=n/e^n\)

17) \(\displaystyle a_k=\frac{k^e}{e^k}\)

Solution: \(\displaystyle (a_k)^{1/k}→1/e<1.\) Converges.

18) \(\displaystyle a_k=\frac{π^k}{k^π}\)

19) \(\displaystyle a_n=(\frac{1}{e}+\frac{1}{n})^n\)

Solution: \(\displaystyle a^{1/n}_n=\frac{1}{e}+\frac{1}{n}→\frac{1}{e}<1.\) Converges.

20) \(\displaystyle a_k=\frac{1}{(1+lnk)^k}\)

21) \(\displaystyle a_n=\frac{(ln(1+lnn))^n}{(lnn)^n}\)

Solution: \(\displaystyle a^{1/n}_n=\frac{(ln(1+lnn))}{(lnn)}→0\) by L’Hôpital’s rule. Converges.

 

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\displaystyle \sum_{k=1}^∞a_k\) with given terms \(\displaystyle a_k\) converges, or state if the test is inconclusive.

22) \(\displaystyle a_k=\frac{k!}{1⋅3⋅5⋯(2k−1)}\)

23) \(\displaystyle a_k=\frac{2⋅4⋅6⋯2k}{(2k)!}\)

Solution: \(\displaystyle \frac{a_{k+1}}{a_k}=\frac{1}{2k+1}→0.\) Converges by ratio test.

24) \(\displaystyle a_k=\frac{1⋅4⋅7⋯(3k−2)}{3^kk!}\)

25) \(\displaystyle a)n=(1−\frac{1}{n})^{n^2}\)

Solution: \(\displaystyle (a_n)^{1/n}→1/e.\) Converges by root test.

26) \(\displaystyle a_k=(\frac{1}{k+1}+\frac{1}{k+2}+⋯+\frac{1}{2k})^k\) (Hint: Compare \(\displaystyle a^{1/k}_k\) to \(\displaystyle ∫^{2k}_k\frac{dt}{t}\).)

27) \(\displaystyle a_k=(\frac{1}{k+1}+\frac{1}{k+2}+⋯+\frac{1}{3k})^k\)

Solution: \(\displaystyle a^{1/k}_k→ln(3)>1.\) Diverges by root test.

28) \(\displaystyle a_n=(n^{1/n}−1)^n\)

 

Use the ratio test to determine whether \(\displaystyle \sum_{n=1}^∞a_n\) converges, or state if the ratio test is inconclusive.

29) \(\displaystyle \sum_{n=1}^∞\frac{3^{n^2}}{2^{n^3}}\)

Solution: \(\displaystyle \frac{a_{n+1}}{a_n}= \frac{3^{2n+1}}{2^{3n^2+3n+1}}→0.\) Converge.

30) \(\displaystyle \sum_{n=1}^∞\frac{2^{n^2}}{n^nn!}\)

 

Use the root and limit comparison tests to determine whether \(\displaystyle \sum_{n=1}^∞a_n\) converges.

31) \(\displaystyle a_n=1/x^n_n\) where \(\displaystyle x_{n+1}=\frac{1}{2}x_n+\frac{1}{x_n}, x_1=1\) (Hint: Find limit of \(\displaystyle {x_n}\).)

Solution: Converges by root test and limit comparison test since \(\displaystyle x_n→\sqrt{2}\).

 

In the following exercises, use an appropriate test to determine whether the series converges.

32) \(\displaystyle \sum_{n=1}^∞\frac{(n+1)}{n^3+n^2+n+1}\)

33) \(\displaystyle \sum_{n=1}^∞\frac{(−1)^{n+1}(n+1)}{n^3+3n^2+3n+1}\)

Solution: Converges absolutely by limit comparison with p−series, \(\displaystyle p=2.\)

34) \(\displaystyle \sum_{n=1}^∞\frac{(n+1)^2}{n^3+(1.1)^n}\)

35) \(\displaystyle \sum_{n=1}^∞\frac{(n−1)^n}{(n+1)^n}\)

Solution: \(\displaystyle \lim_{n→∞}a_n=1/e^2≠0\). Series diverges.

36) \(\displaystyle a_n=(1+\frac{1}{n^2})^n\) (Hint: \(\displaystyle (1+\frac{1}{n^2})^{n^2}≈e.)\)

37) \(\displaystyle a_k=1/2^{sin^2k}\)

Solution: Terms do not tend to zero: \(\displaystyle a_k≥1/2,\) since \(\displaystyle sin^2x≤1.\)

38) \(\displaystyle a_k=2^{−sin(1/k)}\)

39) \(\displaystyle a_n=1/(^{n+2}_n)\) where \(\displaystyle (^n_k)=\frac{n!}{k!(n−k)!}\)

Solution: \(\displaystyle a_n=\frac{2}{(n+1)(n+2)},\) which converges by comparison with p−series for \(\displaystyle p=2\).

40) \(\displaystyle a_k=1/(^{2k}_k)\)

41) \(\displaystyle a_k=2^k/(^{3k}_k)\)

Solution: \(\displaystyle a_k=\frac{2^k1⋅2⋯k}{(2k+1)(2k+2)⋯3k}≤(2/3)^k\) converges by comparison with geometric series.

42) \(\displaystyle a_k=(\frac{k}{k+lnk})^k\) (Hint: \(\displaystyle a_k=(1+\frac{lnk}{k})^{−(k/lnk)lnk}≈e^{−lnk}\).)

43) \(\displaystyle a_k=(\frac{k}{k+lnk})^{2k}\) (Hint: \(\displaystyle a_k=(1+\frac{lnk}{k})^{−(k/lnk)lnk^2}.)\)

Solution: \(\displaystyle a_k≈e^{−lnk^2}=1/k^2.\) Series converges by limit comparison with \(\displaystyle p−series, p=2.\)

 

The following series converge by the ratio test. Use summation by parts, \(\displaystyle \sum_{k=1}^na_k(b_{k+1}−b_k)=[a_{n+1}b_{n+1}−a_1b_1]−\sum_{k=1}^nb_{k+1}(a_{k+1}−a_k),\) to find the sum of the given series.

44) \(\displaystyle \sum_{k=1}^∞\frac{k}{2^k}\) (Hint: Take \(\displaystyle a_k=k\) and \(\displaystyle b_k=2^{1−k}\).)

45) \(\displaystyle \sum_{k=1}^∞\frac{k}{c^k},\) where \(\displaystyle c>1\) (Hint: Take \(\displaystyle a_k=k\) and \(\displaystyle b_k=c^{1−k}/(c−1)\).)

Solution: If \(\displaystyle b_k=c^{1−k}/(c−1)\) and \(\displaystyle a_k=k\), then \(\displaystyle b_{k+1}−b_k=−c^{−k}\) and \(\displaystyle \sum_{n=1}^∞\frac{k}{c^k}=a_1b_1+\frac{1}{c−1}\sum_{k=1}^∞c^{−k}=\frac{c}{(c−1)^2}.\)

46) \(\displaystyle \sum_{n=1}^∞\frac{n^2}{2^n}\)

47) \(\displaystyle \sum_{n=1}^∞\frac{(n+1)^2}{2^n}\)

Solution: \(\displaystyle 6+4+1=11\)

 

The kth term of each of the following series has a factor \(\displaystyle x^k\). Find the range of \(\displaystyle x\) for which the ratio test implies that the series converges.

48) \(\displaystyle \sum_{k=1}^∞\frac{x^k}{k^2}\)

49) \(\displaystyle \sum_{k=1}^∞\frac{x^{2k}}{k^2}\)

Solution: \(\displaystyle |x|≤1\)

50) \(\displaystyle \sum_{k=1}^∞\frac{x^{2k}}{3^k}\)

51) \(\displaystyle \sum_{k=1}^∞\frac{x^k}{k!}\)

Solution: \(\displaystyle |x|<∞\)

52) Does there exist a number \(\displaystyle p\) such that \(\displaystyle \sum_{n=1}^∞\frac{2^n}{n^p}\) converges?

53) Let \(\displaystyle 0<r<1.\) For which real numbers \(\displaystyle p\) does \(\displaystyle \sum_{n=1}^∞n^pr^n\) converge?

Solution: All real numbers \(\displaystyle p\) by the ratio test.

54) Suppose that \(\displaystyle \lim_{n→∞}∣\frac{a_{n+1}}{a_n}∣=p.\) For which values of \(\displaystyle p\) must \(\displaystyle \sum_{n=1}^∞2^na_n\) converge?

55) Suppose that \(\displaystyle \lim_{n→∞}∣\frac{a_{n+1}}{a_n}∣=p.\) For which values of \(\displaystyle r>0\) is \(\displaystyle \sum_{n=1}^∞r^na_n\) guaranteed to converge?

Solution: \(\displaystyle r<1/p\)

56) Suppose that \(\displaystyle ∣\frac{a_{n+1}}{a_n}∣ ≤(n+1)^p\) for all \(\displaystyle n=1,2,…\) where \(\displaystyle p\) is a fixed real number. For which values of \(\displaystyle p\) is \(\displaystyle \sum_{n=1}^∞n!a_n\) guaranteed to converge?

57) For which values of \(\displaystyle r>0\), if any, does \(\displaystyle \sum_{n=1}^∞r^{\sqrt{n}}\) converge? (Hint: \(\displaystyle sum_{n=1}^∞a_n=\sum_{k=1}^∞\sum_{n=k^2}^{(k+1)^2−1}a_n.)\)

Solution: \(\displaystyle 0<r<1.\) Note that the ratio and root tests are inconclusive. Using the hint, there are \(\displaystyle 2k\) terms \(\displaystyle r^\sqrt{n}\) for \(\displaystyle k^2≤n<(k+1)^2\), and for \(\displaystyle r<1\) each term is at least \(\displaystyle r^k\). Thus, \(\displaystyle \sum_{n=1}^∞r^{\sqrt{n}}=\sum_{k=1}^∞\sum_{n=k^2}^{(k+1)^2−1}r^{\sqrt{n}} ≥\sum_{k=1}^∞2kr^k,\) which converges by the ratio test for \(\displaystyle r<1\). For \(\displaystyle r≥1\) the series diverges by the divergence test.

58) Suppose that \(\displaystyle ∣\frac{a_{n+2}}{a_n}∣ ≤r<1\) for all \(\displaystyle n\). Can you conclude that \(\displaystyle \sum_{n=1}^∞a_n\) converges?

59) Let \(\displaystyle a_n=2^{−[n/2]}\) where \(\displaystyle [x]\) is the greatest integer less than or equal to x. Determine whether \(\displaystyle \sum_{n=1}^∞a_n\) converges and justify your answer.

Solution: One has \(\displaystyle a_1=1, a_2=a_3=1/2,…a_{2n}=a_{2n+1}=1/2^n\). The ratio test does not apply because \(\displaystyle a_{n+1}/a_n=1\) if \(\displaystyle n\) is even. However, \(\displaystyle a_{n+2}/a_n=1/2,\) so the series converges according to the previous exercise. Of course, the series is just a duplicated geometric series.

 

The following advanced exercises use a generalized ratio test to determine convergence of some series that arise in particular applications when tests in this chapter, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if \(\displaystyle \lim_{n→∞}\frac{a_{2n}}{a_n}<1/2\), then \(\displaystyle \sum a_n\) converges, while if \(\displaystyle \lim_{n→∞}\frac{a_{2n+1}}{a_n}>1/2\), then \(\displaystyle \sum a_n\) diverges.

60) Let \(\displaystyle a_n=\frac{1}{4}\frac{3}{6}\frac{5}{8}⋯\frac{2n−1}{2n+2}=\frac{1⋅3⋅5⋯(2n−1)}{2^n(n+1)!}\). Explain why the ratio test cannot determine convergence of \(\displaystyle \sum_{n=1}^∞a_n\). Use the fact that \(\displaystyle 1−1/(4k)\) is increasing \(\displaystyle k\) to estimate \(\displaystyle \lim_{n→∞}\frac{a_{2n}}{a_n}\).

61) Let \(\displaystyle a_n=\frac{1}{1+x}\frac{2}{2+x}⋯\frac{n}{n+x}\frac{1}{n}=\frac{(n−1)!}{(1+x)(2+x)⋯(n+x).}\) Show that \(\displaystyle a_{2n}/a_n≤e^{−x/2}/2\). For which \(\displaystyle x>0\) does the generalized ratio test imply convergence of \(\displaystyle \sum_{n=1}^∞a_n\)? (Hint: Write \(\displaystyle 2a_{2n}/a_n\) as a product of \(\displaystyle n\) factors each smaller than \(\displaystyle 1/(1+x/(2n)).)\)

Solution: \(\displaystyle a_{2n}/a_n=\frac{1}{2}⋅\frac{n+1}{n+1+x}\frac{n+2}{n+2+x}⋯\frac{2n}{2n+x}.\) The inverse of the \(\displaystyle kth\) factor is \(\displaystyle (n+k+x)/(n+k)>1+x/(2n)\) so the product is less than \(\displaystyle (1+x/(2n))^{−n}≈e^{−x/2}.\) Thus for \(\displaystyle x>0, \frac{a_{2n}}{a_n}≤\frac{1}{2}e^{−x/2}\). The series converges for \(\displaystyle x>0\).

62) Let \(\displaystyle a_n=\frac{n^{lnn}}{(lnn)^n}.\) Show that \(\displaystyle \frac{a_{2n}}{a_n}→0\) as \(\displaystyle n→∞.\)

Chapter Review Exercise

True or False? Justify your answer with a proof or a counterexample.

1) If \(\displaystyle \lim_{n→∞}a_n=0,\) then \(\displaystyle \sum_{n=1}^∞a_n\) converges.

Solution: false

2) If \(\displaystyle \lim_{n→∞}a_n≠0,\) then \(\displaystyle \sum_{n=1}^∞a_n\) diverges.

3) If \(\displaystyle \sum_{n=1}^∞|a_n|\) converges, then \(\displaystyle \sum_{n=1}^∞a_n\) converges.

Solution: true

4) If \(\displaystyle \sum_{n=1}^∞2^na_n\) converges, then \(\displaystyle \sum_{n=1}^∞(−2)^na_n\) converges.

 

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit.

5) \(\displaystyle a_n=\frac{3+n^2}{1−n}\)

Solution: unbounded, not monotone, divergent

6) \(\displaystyle a_n=ln(\frac{1}{n})\)

7) \(\displaystyle a_n=\frac{ln(n+1)}{\sqrt{n+1}}\)

Solution: bounded, monotone, convergent, \(\displaystyle 0\)

8) \(\displaystyle a_n=\frac{2^{n+1}}{5^n}\)

9) \(\displaystyle a_n=\frac{ln(cosn)}{n}\)

Solution: unbounded, not monotone, divergent

 

Is the series convergent or divergent?

10) \(\displaystyle \sum_{n=1}^∞\frac{1}{n^2+5n+4}\)

11) \(\displaystyle \sum_{n=1}^∞ln(\frac{n+1}{n})\)

Solution: diverges

12) \(\displaystyle \sum_{n=1}^∞\frac{2^n}{n^4}\)

13) \(\displaystyle \sum_{n=1}^∞\frac{e^n}{n!}\)

Solution: converges

14) \(\displaystyle \sum_{n=1}^∞n^{−(n+1/n)}\)

 

Is the series convergent or divergent? If convergent, is it absolutely convergent?

15) \(\displaystyle \sum_{n=1}^∞\frac{(−1)^n}{\sqrt{n}}\)

Solution: converges, but not absolutely

16) \(\displaystyle \sum_{n=1}^∞\frac{(−1)^nn!}{3^n}\)

17) \(\displaystyle \sum_{n=1}^∞\frac{(−1)^nn!}{n^n}\)

Solution: converges absolutely

18) \(\displaystyle \sum_{n=1}^∞sin(\frac{nπ}{2})\)

19) \(\displaystyle \sum_{n=1}^∞cos(πn)e^{−n}\)

Solution: converges absolutely

 

Evaluate

20) \(\displaystyle \sum_{n=1}^∞\frac{2^{n+4}}{7^n}\)

21) \(\displaystyle \sum_{n=1}^∞\frac{1}{(n+1)(n+2)}\)

Solution: \(\displaystyle \frac{1}{2}\)

22) A legend from India tells that a mathematician invented chess for a king. The king enjoyed the game so much he allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, and so on. Find an exact expression for the total payment (in grains of rice) requested by the mathematician. Assuming there are \(\displaystyle 30,000\) grains of rice in \(\displaystyle 1\) pound, and \(\displaystyle 2000\) pounds in \(\displaystyle 1\) ton, how many tons of rice did the mathematician attempt to receive?

 

The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula \(\displaystyle x_{n+1}=bx_n\), where \(\displaystyle x_n\) is the population of houseflies at generation \(\displaystyle n\), and \(\displaystyle b\) is the average number of offspring per housefly who survive to the next generation. Assume a starting population \(\displaystyle x_0\).

23) Find \(\displaystyle \lim_{n→∞}x_n\) if \(\displaystyle b>1, b<1\), and \(\displaystyle b=1.\)

Solution: \(\displaystyle ∞, 0, x_0\)

24) Find an expression for \(\displaystyle S_n=\sum_{i=0}^nx_i\) in terms of \(\displaystyle b\) and \(\displaystyle x_0\). What does it physically represent?

25) If \(\displaystyle b=\frac{3}{4}\) and \(\displaystyle x_0=100\), find \(\displaystyle S_{10}\) and \(\displaystyle \lim_{n→∞}S_n\)

Solution: \(\displaystyle S_{10}≈383, \lim_{n→∞}S_n=400\)

26) For what values of \(\displaystyle b\) will the series converge and diverge? What does the series converge to?