91Ó°ÊÓ

Determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{\ln n}{n}$$

Find the binomial series for the functions. $$(1+x)^{4}$$

Find the Maclaurin serics for the functions in Exercises \(11-24\) $$e^{-x}$$

Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right)$$

Use the Integral Test to determine whether the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$\sum_{n=1}^{\infty} \frac{7}{\sqrt{n+4}}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely,(c) conditionally? $$\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n}}{n !}$$

Use the limit Comparison Test to determine whether each series converges or diverges. $$\sum_{n=2}^{\infty} \frac{n(n+1)}{\left(n^{2}+1\right)(n-1)}$$

Use the Root Test to determine whether each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}\left(\frac{4 n+3}{3 n-5}\right)^{n}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely,(c) conditionally? $$\sum_{n=0}^{\infty} \frac{3^{n} x^{n}}{n !}$$

Use the Root Test to determine whether each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}\left(-\ln \left(e^{2}+\frac{1}{n}\right)\right)^{n+1}$$