91Ó°ÊÓ

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{x}{9+x^{2}} $$

Use the Integral Test to determine whether the series is convergent or divergent. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt[5]{n}}\)

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \(10-2+0.4-0.08+\cdots\)

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=0}^{\infty} \frac{x^{n}}{n !}$$

\(3-8=\) Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty}(-1)^{n} \cos \left(\frac{\pi}{n}\right) $$

\(5-8=\) Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues. $$\\{5,8,11,14,17, \ldots\\}$$

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \(2+0.5+0.125+0.03125+\cdots\)

Find the Maclaurin series for \(f(x)\) using the definition of a Maclaurin series. [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0 . ]\) Also find the associated radius of convergence. $$f(x)=x \cos x$$

Find a power series representation for the function and determine the interval of convergence. $$ f(x)=\frac{x}{2 x^{2}+1} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n+4}}$$