$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# Chapter 9 Review Exercises

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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.

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.

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) $$a_n=\dfrac{3+n^2}{1−n}$$

unbounded, not monotone, divergent

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

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

bounded, monotone, convergent, $$0$$

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

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

unbounded, not monotone, divergent

Is the series convergent or divergent?

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

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

diverges

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

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

converges

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

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

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

converges, but not absolutely

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

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

converges absolutely

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

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

converges absolutely

Evaluate.

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

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

$$\frac{1}{2}$$

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 $$b=1.$$

$$∞, \; 0, \; x_0$$

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 $$\displaystyle \lim_{n→∞}S_n$$

$$\displaystyle S_{10}≈383, \quad \lim_{n→∞}S_n=400$$
26) For what values of $$b$$ will the series converge and diverge? What does the series converge to?