8E: Chapter Review Excercises
This page is a draft and is under active development.
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Exercise 8E.1 True or False?
True or False? Justify your answer with a proof or a counterexample.
1) If limn→∞an=0, then ∞∑n=1an converges.
- Answer
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Solution: false
2) If limn→∞an≠0, then ∞∑n=1an diverges.
3) If ∞∑n=1|an| converges, then ∞∑n=1an converges.
- Answer
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Solution: true
4) If ∞∑n=12nan converges, then ∞∑n=1(−2)nan converges.
Exercise 8E.2 bounded, monotone, and convergent or divergent?
Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit.
1) an=3+n21−n
- Answer
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Solution: unbounded, not monotone, divergence
2) an=ln(1n)
3) an=ln(n+1)√n+1
- Answer
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Solution: bounded, monotone, convergent, 0
4) an=2n+15n
5) an=ln(cosn)n
- Answer
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Solution: unbounded, not monotone, divergent
Exercise 8E.3 convergent or divergent?
Is the series convergent or divergent?
1) ∞∑n=11n2+5n+4
2) ∞∑n=1ln(n+1n)
- Answer
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Solution: diverges
3) ∞∑n=12nn4
4) ∞∑n=1enn!
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Solution: converge
5) ∞∑n=1n−(n+1/n)
Exercise 8E.4 convergent or divergent? If convergent, is it absolutely convergent?
Is the series convergent or divergent? If convergent, is it absolutely convergent?
1) ∞∑n=1(−1)n√n
- Answer
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Solution: converges, but not absolutely.
2) ∞∑n=1(−1)nn!3n
3) ∞∑n=1(−1)nn!nn
- Answer
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Solution: converges absolutel
4) \displaystyle \sum_{n=1}^∞sin(\frac{nπ}{2})
5) \displaystyle \sum_{n=1}^∞cos(πn)e^{−n}
- Answer
- Solution: converges absolutely
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Exercise \PageIndex{5} Evaluate
Evaluate
1) \displaystyle \sum_{n=1}^∞\frac{2^{n+4}}{7^n}
2) \displaystyle \sum_{n=1}^∞\frac{1}{(n+1)(n+2)}
- Answer
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Solution: \displaystyle \frac{1}{2}
Exercise \PageIndex{6}
1) 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?
Exercise \PageIndex{7}
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.
1) Find \displaystyle \lim_{n→∞}x_n if \displaystyle b>1, b<1, and \displaystyle b=1.
- Answer
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Solution: \displaystyle ∞, 0, x_0
2) 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?
3) If \displaystyle b=\frac{3}{4} and \displaystyle x_0=100, find \displaystyle S_{10} and \displaystyle \lim_{n→∞}S_n
- Answer
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Solution: \displaystyle S_{10}≈383, \lim_{n→∞}S_n=400
4) For what values of \displaystyle b will the series converge and diverge? What does the series converge to?