9.R: Chapter 9 Review Exercises

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.

Answer

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.

Answer

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) $a_n=\dfrac{3+n^2}{1−n}$

Answer

unbounded, not monotone, divergent

6) $a_n=\ln\left(\frac{1}{n}\right)$

7) $a_n=\dfrac{\ln(n+1)}{\sqrt{n+1}}$

Answer

bounded, monotone, convergent, $0$

8) $a_n=\dfrac{2^{n+1}}{5^n}$

9) $a_n=\dfrac{\ln(\cos n)}{n}$

Answer

unbounded, not monotone, divergent

Is the series convergent or divergent?

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

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

Answer

diverges

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

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

Answer

converges

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

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

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

Answer

converges, but not absolutely

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

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

Answer

converges absolutely

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

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

Answer

converges absolutely

Evaluate.

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

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

Answer

$\frac{1}{2}$

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 $30,000$ grains of rice in $1$ pound, and $2000$ pounds in $1$ 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 $x_{n+1}=bx_n$, where $x_n$ is the population of houseflies at generation $n$, and $b$ is the average number of offspring per housefly who survive to the next generation. Assume a starting population $x_0$.

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

Answer

$∞, \; 0, \; x_0$

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

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

Answer

$\displaystyle S_{10}≈383, \quad \lim_{n→∞}S_n=400$

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