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9.7: Chapter 9 Review Exercises

  • Page ID
    137800
    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax

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    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 9.7: 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; a detailed edit history is available upon request.