91Ó°ÊÓ

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

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

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^{n}}$$

Use the Comparison Test to determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \frac{n}{2 n^{3}+1}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=1-(0.2)^{n}$$

\(9-12=\) Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{6}}( | \text { error } |<0.00005) $$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{n^{2} x^{n}}{2^{n}}$$

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. $$f(x)=\sinh x$$

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number \(a\) . (b) Use Taylor's Formula to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=\sqrt{x}, \quad a=4, \quad n=2, \quad 4 \leqslant x \leqslant 4.2$$

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