
# 9.4E: Exercises for Comparison Test

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Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges.

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

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

Converges by comparison with $$\dfrac{1}{n^2}$$.

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

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

Diverges by comparison with harmonic series, since $$2n−1≥n.$$

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

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

$$a_n=1/(n+1)(n+2)<1/n^2.$$ Converges by comparison with $$p$$-series, $$p=2>1$$.

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

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

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

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

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

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

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

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

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

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

Use the Limit Comparison Test to determine whether each series in exercises 14 - 28 converges or diverges.

14) $$\displaystyle \sum^∞_{n=1}\left(\frac{\ln n}{n}\right)^2$$

Converges by limit comparison with $$p$$-series for $$p>1$$.

15) $$\displaystyle \sum^∞_{n=1}\left(\frac{\ln n}{n^{0.6}}\right)^2$$

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

Converges by limit comparison with $$p$$-series, $$p=2>1.$$

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

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

Converges by limit comparison with $$4^{−n}$$.

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

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

Converges by limit comparison with $$1/e^{1.1n}$$.

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

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

Diverges by limit comparison with harmonic series.

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

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

Converges by limit comparison with $$p$$-series, $$p=3>1$$.

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

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

Converges by limit comparison with $$p$$-series, $$p=3>1$$.

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

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

Diverges by limit comparison with $$1/n$$.

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

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

Converges for $$p>1$$ by comparison with a $$p$$ series for slightly smaller $$p$$.

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

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

Converges for all $$p>0$$.

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

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

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

35) Find all values of $$p$$ and $$q$$ 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.

The numerator is equal to $$1$$ when $$n$$ is odd and $$0$$ when $$n$$ 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 $$n$$, at least one of $${|\sin n|,|\sin(n+1)|,...,|\sin(n+6)|}$$ is larger than $$1/2$$. Use this relation to test convergence of $$\displaystyle \sum^∞_{n=1}\frac{|\sin n|}{\sqrt{n}}$$.

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

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

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

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

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

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

42) Show that if $$a_n≥0$$ 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?

$$a_n→0,$$ so $$a^2_n≤|a_n|$$ for large $$n$$. Convergence follows from limit comparison. $$\displaystyle \sum_{n=1}^∞\frac{1}{n^2}$$ converges, but $$\displaystyle \sum_{n=1}^∞\frac{1}{n}$$ 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 $$a_n>0$$ for all $$n$$ and that $$\displaystyle \sum_{n=1}^∞a_n$$ converges. Suppose that $$b_n$$ is an arbitrary sequence of zeros and ones. Does $$\displaystyle \sum_{n=1}^∞a_nb_n$$ necessarily converge?

44) Suppose that $$a_n>0$$ for all $$n$$ and that $$\displaystyle \sum_{n=1}^∞a_n$$ diverges. Suppose that $$b_n$$ 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?

No. $$\displaystyle \sum_{n=1}^∞1/n$$ diverges. Let $$b_k=0$$ unless $$k=n^2$$ for some $$n$$. 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 $$s$$, then $$\dfrac{1}{2}s=\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6}+⋯$$ and $$s−\dfrac{1}{2}s=1+\dfrac{1}{3}+\dfrac{1}{5}+⋯.$$ Why does this lead to a contradiction?

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

$$|\sin t|≤|t|,$$ so the result follows from the comparison test.

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

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

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

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

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

If $$b_1=0,$$ 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 $$1$$-kg weights, and one each of $$1/2$$-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 $$1$$-kg weights, and nine each of $$0.1$$-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?

Yes. Keep adding $$1$$-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 $$1$$-kg weights, and add $$0.1$$-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 \$$0.1$$-kg weight. Start adding $$0.01$$-kg weights. If it balances, stop. If it tips to the side with the weights, remove the last $$0.01$$-kg weight that was added. Continue in this way for the $$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$$ where $$A$$ is the number of full kg weights and $$d_n$$ is the number of $$1/10^n$$-kg weights that were added. If at some state this series is Robert’s exact weight, the process will stop. Otherwise it represents the $$N^{\text{th}}$$ partial sum of an infinite series that gives Robert’s exact weight, and the error of this sum is at most $$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 $$n$$ is odd. Let $$m>1$$ be fixed. Show, more generally, that deleting all terms $$1/n$$ where $$n=mk$$ for some integer $$k$$ 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 $$1/n$$ if a given digit, say $$9$$, appears in the decimal expansion of $$n$$. Argue that this depleted harmonic series converges by answering the following questions.

a. How many whole numbers $$n$$ have $$d$$ digits?

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

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

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.

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

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

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

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