
8E: Chapter Review Excercises

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Exercise $$\PageIndex{1}$$ True or False?

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.

Exercise $$\PageIndex{2}$$ bounded, monotone, and convergent or divergent?

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit.

1) $$\displaystyle a_n=\frac{3+n^2}{1−n}$$

Solution: unbounded, not monotone, divergence

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

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

Solution: bounded, monotone, convergent, $$\displaystyle 0$$

4) $$\displaystyle a_n=\frac{2^{n+1}}{5^n}$$

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

Solution: unbounded, not monotone, divergent

Exercise $$\PageIndex{3}$$ convergent or divergent?

Is the series convergent or divergent?

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

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

Solution: diverges

3) $$\displaystyle \sum_{n=1}^∞\frac{2^n}{n^4}$$

4) $$\displaystyle \sum_{n=1}^∞\frac{e^n}{n!}$$

Solution: converge

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

Exercise $$\PageIndex{4}$$ convergent or divergent? If convergent, is it absolutely convergent?

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

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

Solution: converges, but not absolutely.

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

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

Solution: converges absolutel

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

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

Solution: converges absolutely

Exercise $$\PageIndex{5}$$ Evaluate

Evaluate

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

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

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 $$\displaystyle b=1.$$

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 $$\displaystyle \lim_{n→∞}S_n$$
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?