Math 132: Calculus for Physical and Life Sciences 2 Problem Set 8 Due Friday, April 11, 2008, at the beginning of class.

General Directions: You must show all work for credit on these problems.

1. Determine whether each of the following sequences converges or diverges. If it converges, find its limit.

(a) a_{n} = \frac{1 − n^{2}}{2 + 3n^{2}}  (b) a_{n} = 1 + (−1)^{n}

(c) a_{n} = \frac{sin n}{3^{n}}  (d) a_{n} = \frac{(ln n)^{2}}{n}

(e) a_{n} = \frac{2^{n} + 1}{e^{n}}  (f) a_{n} = ln(2n + 1) − ln(2n − 1)

2. Determine if each infinite series converges or diverges. If it converges, find its sum. (Be careful about the first term of the series.)

(a) 1 + e^{−1} + e^{−2} + e^{−3} + … + e^{−n} + …

\begin{matrix}(b) ∑_{n=0}^{∞} \frac{3^{n} − 2^{n}}{4^{n}} & (c) ∑_{n=0}^{∞} (\frac{100}{99})^{n} & (d) ∑_{n=0}^{∞} (\frac{99}{100})^{n} \\ (e) ∑_{n=1}^{∞} (\frac{π}{e} )^{n} & (f) ∑_{n=1}^{∞} (\frac{2}{n} − \frac{1}{2^{n}} ) & (g) ∑_{k=1}^{∞} (\frac{k + 1}{2k + 4}) \end{matrix}

(h) ∑_{n=1}^{∞} ln (\frac{n + 1}{n})

(Hint for part (h): Consider several partial sums and use properties of logarithms to simplify them.)

3. Find the rational number represented by 0.2525252525 . . ..

(Hint: Think of 0.2525252525 … as a geometric series.)

4. Suppose the government spends $1 billion and that each recipient spends 90% of the dollars that he or she receives. In turn, the secondary recipients spend 90% of the dollars they receive, and so on. How much total spending results from the original injection of $1 billion into the country?

5. Use the integral test to decide whether each of the following series converges or diverges.

(a) ∑_{n=0}^{∞} \frac{n}{n^{2} + 1}  (b) ∑_{n=0}^{∞} \frac{n}{e^{n^{2}}}  (c) ∑_{n=1}^{∞} \frac{1}{n ln n}  (d) ∑_{n=1}^{∞} \frac{n}{e^{n}}

(e) ∑_{n=1}^{∞} ln (1 + \frac{1}{n^{2}} )

(Hint for part (e): integrate by parts.)

6. Determine whether the given series converges or diverges. Quote general results to justify your answers.

(a) ∑_{n=1}^{∞} \frac{1}{(\sqrt{n})^{5}} (b) ∑_{n=1}^{∞} \frac{n + 1}{n^{2}}

7. Show that the series ∑_{n=2}^{∞} \frac{1}{n(ln n)^{p}} converges if p > 1 and diverges if p ≤ 1.

8. For each of the following series explain why the integral test does not apply.

(a) ∑_{n=1}^{∞} e^{−n} sin n (b) ∑_{n=1}^{∞} \frac{2 + sin n}{n^{2}}

Selected answers:

1. (a) sequence converges to −\frac{1}{3}, (c) sequence converges to 0, (f) sequence converges to 0.

2. (b) series converges and the sum is 2, (f) series diverges, (h) series diverges.

3. \frac{25}{99}

5. (b) series converges, (d) series converges.

6. (a) series converges.