Section 11.6

Key Problems: 1, 2, 4, 7-9, 11, 19, 21, 29, 31, 41

Problem 1

What can you say about the series $\sum a_n$ in each of the following cases?

(a) ${\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|=8$

(b) ${\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|=0.8$

(c) ${\lim }_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|=1$

Problem 2

Suppose that for the series $\sum a_n$ we have ${\lim }_{n\to \infty }|a_n/a_{n+1}|=2$. What is ${\lim }_{n\to \infty }|a_{n+1}/a_n|$? Does the series converge?

Problem 4

Use the Ratio Test to determine whether the series is convergent or divergent.

$$\sum _{n=1}^{\infty }\frac{(-2{)}^n}{n^2}$$

Problem 7

$$\sum _{k=1}^{\infty }\frac{1}{k!}$$

Problem 8

$$\sum _{k=1}^{\infty }k{e}^{-k}$$

Problem 9

$$\sum _{n=1}^{\infty }\frac{10^n}{(n+1)4^{2n+1}}$$

Problem 11

$$\sum _{n=1}^{\infty }\frac{n{\pi }^n}{(-3{)}^{n-1}}$$

Problem 19¹

$$\sum _{n=1}^{\infty }\frac{2\times 4\times 6\times \cdots \times (2n)}{n!}$$

Problem 21

Use the Root Test to determine whether the series is convergent or divergent.

$$\sum _{n=1}^{\infty }{\left(\frac{n^2+1}{2n^2+1}\right)}^n$$

Problem 29

Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

$$\sum _{n=1}^{\infty }\frac{(-9{)}^n}{n10^{n+1}}$$

Problem 31

$$\sum _{n=2}^{\infty }{\left(\frac{n}{\ln n}\right)}^n$$

Problem 41

(a) Show that ${\sum }_{n=0}^{\infty }{x}^n/n!$ converges for all $x$.

(b) Deduce that ${\lim }_{n\to \infty }{x}^n/n!=0$ for all $x$.

Footnotes

1. $\times$ is “times” not the cross-product ↩