9.R: Chapter 9 Review Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
True or False? Justify your answer with a proof or a counterexample.
1) If limn→∞an=0, then ∞∑n=1an converges.
- Answer
- false
2) If limn→∞an≠0, then ∞∑n=1an diverges.
3) If ∞∑n=1|an| converges, then ∞∑n=1an converges.
- Answer
- true
4) If ∞∑n=12nan converges, then ∞∑n=1(−2)nan converges.
Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit.
5) an=3+n21−n
- Answer
- unbounded, not monotone, divergent
6) an=ln(1n)
7) an=ln(n+1)√n+1
- Answer
- bounded, monotone, convergent, 0
8) an=2n+15n
9) an=ln(cosn)n
- Answer
- unbounded, not monotone, divergent
Is the series convergent or divergent?
10) ∞∑n=11n2+5n+4
11) ∞∑n=1ln(n+1n)
- Answer
- diverges
12) ∞∑n=12nn4
13) ∞∑n=1enn!
- Answer
- converges
14) ∞∑n=1n−(n+1/n)
Is the series convergent or divergent? If convergent, is it absolutely convergent?
15) ∞∑n=1(−1)n√n
- Answer
- converges, but not absolutely
16) ∞∑n=1(−1)nn!3n
17) ∞∑n=1(−1)nn!nn
- 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?