91Ó°ÊÓ

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=2}^{\infty} \frac{x^{n}}{n(\ln n)^{2}}$$

Which of the series in Exercises converge, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=2}^{\infty} \frac{1}{\sqrt{n} \ln n}$$

Use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=0}^{\infty} \frac{1}{n+4}$$

Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{\ln n}{n-\ln n}$$

Use the \(n\) th-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. $$\sum_{n=1}^{\infty} \frac{n}{n^{2}+3}$$

Find the Taylor series generated by \(f\) at \(x=a.\) $$f(x)=2^{x}, \quad a=1$$

Use series to evaluate the limits. $$\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{x}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=2}^{\infty} \frac{x^{n}}{n \ln n}$$

Find the first four nonzero terms in the Maclaurin series for the functions. $$\frac{\ln (1+x)}{1-x}$$

Which of the series in Exercises converge, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(\ln n)^{2}}{n^{3 / 2}}$$