Section 11.2

Key Problems: 14, 15, 37-42 all, 54, 61, 63

Problem 14

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.

$$\sum _{n=1}^{\infty }\frac{7^{n+1}}{10^n}$$

Problem 15

Let $a_n=\frac{2n}{3n+1}$.

(a) Determine whether $\{a_n\}$ is convergent.

(b) Determine whether ${\sum }_{n=1}^{\infty }{a}_n$ is convergent.

Problem 37

Determine whether the series is convergent or divergent. If it is convergent, find its sum.

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

Problem 38

$$\sum _{k=1}^{\infty }\frac{k^2}{k^2-2k+5}$$

Problem 39

$$\sum _{n=1}^{\infty }{3}^{n+1}{4}^{-n}$$

Problem 40

$$\sum _{n=1}^{\infty }[(-0.2{)}^n+(0.6{)}^{n-1}]$$

Problem 41

$$\sum _{n=1}^{\infty }\frac{1}{4+e^{-n}}$$

Problem 42

$$\sum _{n=1}^{\infty }\frac{2^n+4^n}{e^n}$$

Problem 54

Express the number as a ratio of integers.

$$0.\overline{46}=0.46464646\dots$$

Problem 61

Find the values of $x$ for which the series converges. Find the sum of the series for those values of $x$.

$$\sum _{n=1}^{\infty }\frac{(x-2{)}^n}{3^n}$$

Problem 63

$$\sum _{n=1}^{\infty }\frac{2^n}{x^n}$$