91Ó°ÊÓ

Each of Exercises \(7-12\) gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. $$ a_{1}=-2, \quad a_{n+1}=n a_{n} /(n+1) $$

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. $$ \sum_{n=0}^{\infty}(-1)^{n} \frac{5}{4^{n}} $$

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. $$ \sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right) $$

Use the Limit Comparison Test to determine if each series converges or diverges. \begin{equation}\sum_{n=2}^{\infty} \frac{n(n+1)}{\left(n^{2}+1\right)(n-1)}\end{equation}

Each of Exercises \(7-12\) gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. $$ a_{1}=a_{2}=1, \quad a_{n+2}=a_{n+1}+a_{n} $$

Find the Maclaurin series for the functions \(e^{-x}\)

Find the binomial series for the functions in Exercises \(11-14\) \begin{equation} (1+x)^{4} \end{equation}

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

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? $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n}}{n !} $$

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}\left(\frac{4 n+3}{3 n-5}\right)^{n}$$