91Ó°ÊÓ

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=2}^{\infty}(-1)^{n} \frac{x^{n}}{4^{n} \ln n}$$

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=0}^{\infty} \frac{\pi^{n}}{3^{n+1}}$$

Find the Taylor series for \(f(x)\) centered at the given value of \(a\) . [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1]\) $$f(x)=x^{4}-3 x^{2}+1, \quad a=1$$

(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 \(\left|R_{n}(x)\right|\) $$f(x)=x^{2 / 3}, \quad a=1, \quad n=3, \quad 0.8 \leqslant x \leqslant 1.2$$

Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence. $$ f(x)=\frac{x+2}{2 x^{2}-x-1} $$

Determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty}\left(n^{-1.4}+3 n^{-1.2}\right)$$

Find the Taylor series for \(f(x)\) centered at the given value of \(a\) . [Assume that \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0.1]\) $$f(x)=x-x^{3}, \quad a=-2$$

(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 \(\left|R_{n}(x)\right|\) $$f(x)=\sin x, \quad a=\pi / 6, \quad n=4, \quad 0 \leqslant x \leqslant \pi / 3$$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty} \frac{x^{n}}{5^{n} n^{5}}$$

\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\frac{n^{3}}{n+1}$$