91Ó°ÊÓ

Find the Taylor series in \(x-a\) through the term \((x-a)^{3}\). $$ \tan x, a=\frac{\pi}{4} $$

Determine convergence or divergence for each of the series. Indicate the test you use. $$\text { Hint: } a_{n}=\frac{1}{n(n+1)}$$ $$ \frac{1}{1^{2}+1}+\frac{2}{2^{2}+1}+\frac{3}{3^{2}+1}+\frac{4}{4^{2}+1}+\cdots $$

Evaluate \(\sum_{k=0}^{\infty}(-1)^{k} x^{k},-1

Indicate whether the given series converges or diverges and give a reason for your conclusion. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\sqrt[n]{3}}$$

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n-1}{n}\)

Find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. \(\frac{x}{2}+\frac{2 x^{2}}{3}+\frac{3 x^{3}}{4}+\frac{4 x^{4}}{5}+\frac{5 x^{5}}{6}+\cdots\)

Determine convergence or divergence for each of the series. Indicate the test you use. $$\text { Hint: } a_{n}=\frac{1}{n(n+1)}$$ $$ \frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\frac{4}{3^{4}}+\cdots $$

Find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. \(\frac{x-1}{1}+\frac{(x-1)^{2}}{2}+\frac{(x-1)^{3}}{3}+\frac{(x-1)^{4}}{4}+\cdots\)

Classify each series as absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{\cos n \pi}{n}\)

In Problems \(21-30,\) find an explicit formula \(a_{n}=\ldots\) for each sequence, determine whether the sequence converges or diverges, and, if it converges, find \(\lim _{n \rightarrow \infty} a_{n}\). $$ -1, \frac{2}{3},-\frac{3}{5}, \frac{4}{7},-\frac{5}{9}, \ldots $$