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Mathematics LibreTexts

8E:

  • Page ID
    18594
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    Chapter Review Exercise

    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.

    Solution: 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.

    Solution: 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) \(\displaystyle a_n=\frac{3+n^2}{1−n}\)

    Solution: unbounded, not monotone, divergent

    6) \(\displaystyle a_n=ln(\frac{1}{n})\)

    7) \(\displaystyle a_n=\frac{ln(n+1)}{\sqrt{n+1}}\)

    Solution: bounded, monotone, convergent, \(\displaystyle 0\)

    8) \(\displaystyle a_n=\frac{2^{n+1}}{5^n}\)

    9) \(\displaystyle a_n=\frac{ln(cosn)}{n}\)

    Solution: unbounded, not monotone, divergent

    Is the series convergent or divergent?

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

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

    Solution: diverges

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

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

    Solution: converges

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

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

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

    Solution: converges, but not absolutely

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

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

    Solution: converges absolutely

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

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

    Solution: converges absolutely

    Evaluate

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

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

    Solution: \(\displaystyle \frac{1}{2}\)

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

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

    Solution: \(\displaystyle ∞, 0, x_0\)

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

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

    Solution: \(\displaystyle S_{10}≈383, \lim_{n→∞}S_n=400\)

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