91Ó°ÊÓ

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

In Exercises \(13-26,\) find a formula for the \(n\) th term of the sequence. The sequence \(1,-1,1,-1,1, \ldots\)

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{-8}{(3+(1 / n))^{2 n}}$$

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\frac{x^{2}}{2}-1+\cos x$$

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=1}^{\infty} \frac{5^{n}}{\sqrt{n} 4^{n}}\end{equation}

In Exercises \(1-36\) , (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$ \sum_{n=1}^{\infty} \frac{4^{n} x^{2 n}}{n} $$

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{n}{n+1} $$

Find the Maclaurin series for the functions \(\frac{1}{1+x}\)

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

In Exercises \(7-14,\) 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. $$ \sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}+\frac{(-1)^{n}}{5^{n}}\right) $$