91Ó°ÊÓ

\(2-20\) Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty} \frac{\sin (n \pi / 2)}{n !} $$

\(2-28\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$\sum_{n=1}^{\infty} \frac{3-\cos n}{n^{2 / 3}-2}$$

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=0}^{\infty}(-1)^{n} \frac{(x-3)^{n}}{2 n+1}$$

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number a. (b) Use Taylor's Inequality 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 Taylor series for \(f(x)\) centered at the given value of a. [Assume that \(f\) hat \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0 . ]\) \(f(x)=1 / x, \quad a=-3\)

\(3-32\) Determine whether the series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3}+1}}$$

\(11-20\) Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. $$\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{4^{n}}$$

\(2-28\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$\sum_{n=2}^{\infty} \frac{(-1)^{n}}{\ln n}$$

Find the Taylor series for \(f(x)\) centered at the given value of a. [Assume that \(f\) hat \(f\) has a power series expansion. Do not show that \(R_{n}(x) \rightarrow 0 . ]\) \(f(x)=\cos x, \quad a=\pi\)

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