91Ó°ÊÓ

\(3-32\) Determine whether the series converges or diverges. $$\sum_{n=1}^{\infty} \frac{\arctan n}{n^{1.2}}$$

\(11-20\) Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. $$3-4+\frac{16}{3}-\frac{64}{9}+\cdots$$

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

Determine whether the series is convergent or divergent. $$\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{14}+\frac{1}{17}+\cdots$$

(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 \(\left|R_{n}(x)\right|\) $$f(x)=x^{-2}, \quad a=1, \quad n=2, \quad 0.9 \leqslant x \leqslant 1.1$$

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

(a) Find a power series representation for \(f(x)=\ln (1+x)\) . What is the radius of convergence? (b) Use part (a) to find a power series for \(f(x)=x \ln (1+x)\) (c) Use part (a) to find a power series for \(f(x)=\ln \left(x^{2}+1\right)\)

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 \(R_{n}(x) \rightarrow 0 . ]\) \(f(x)=x-x^{3}, \quad a=-2\)

Test the series for convergence or divergence. $$\sum_{n=1}^{\infty} \frac{\sin 2 n}{1+2^{n}}$$

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