91Ó°ÊÓ

Find the first four terms of the Taylor series for the functions. \(\left(1+\frac{x}{2}\right)^{-2}\)

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{n}{n^{2}+1}$$

Converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$\sum_{n=1}^{\infty} e^{-2 n}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=0}^{\infty} \frac{(x-2)^{n}}{10^{n}}$$

Use the Ratio Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{n^{4}}{(-4)^{n}}$$

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\dots+\frac{1}{(n+1)(n+2)}+\cdots$$

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n^{2}+5}{n^{2}+4}$$

Use the Ratio Test to determine if each series converges absolutely or diverges. $$\sum_{n=2}^{\infty} \frac{3^{n+2}}{\ln n}$$

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$\frac{5}{1 \cdot 2}+\frac{5}{2 \cdot 3}+\frac{5}{3 \cdot 4}+\dots+\frac{5}{n(n+1)}+\cdots$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=0}^{\infty}(2 x)^{n}$$