91Ó°ÊÓ

Which of the series converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n !}{2^{n}}$$

Find the Maclaurin serics for the functions in Exercises \(11-24\) $$\sinh x=\frac{e^{x}-e^{-x}}{2}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely,(c) conditionally? $$\sum_{n=1}^{\infty} \sqrt[n]{n(2 x+5)^{n}}$$

Use series to approximate the values of the integrals in Exercises \(19-\) 22 with an error of magnitude less than \(10^{-8}\) $$\int_{0}^{0.1} e^{-x^{2}} d x$$

Converge, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{1+\cos n}{n^{2}}$$

Determine whether the geometric series converges or diverges. If a series converges, find its sum. $$\left(\frac{1}{3}\right)^{-2}-\left(\frac{1}{3}\right)^{-1}+1-\left(\frac{1}{3}\right)+\left(\frac{1}{3}\right)^{2}-\cdots$$

Find a formula for the \(n\) th term of the sequence. \(-3,-2,-1,0,1, \ldots \quad\) Integers, beginning with -3

Converge, and which diverge? Use any method, and give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{2 n}{3 n-1}$$

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

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