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Mathematics LibreTexts

9.R: Chapter 9 Review Exercises

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

( \newcommand{\kernel}{\mathrm{null}\,}\)

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?


This page titled 9.R: Chapter 9 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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