9.R: Chapter 9 Review Exercises
- Page ID
- 72439
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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?