91Ó°ÊÓ

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-2)^{n-1}}{n^{2}}\)

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2}} $$

Find the \(n\) th partial sum \(S_{n}\) of the telescoping series, and use it to determine whether the series converges or diverges. If it converges, find its sum. \(\sum_{n=1}^{\infty}\left(\frac{-8}{4 n^{2}+4 n-3}\right)\)

Determine whether the series is convergent, absolutely convergent, conditionally convergent, or divergent. \(\sum_{n=1}^{\infty} \frac{(-2)^{n}}{n !}\)

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\frac{1 \cdot 3 \cdot 5 \cdots \cdot(2 n-1)}{n !}\)

Use Equation (1) to find the Taylor series of \(f\) at the given value of \(c .\) Then find the radius of convergence of the series. \(f(x)=e^{-2 x}, \quad c=3\)

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=2}^{\infty} \frac{1}{n^{2 / 3}-1}\)

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n-1} n^{2}}{2 n^{2}-1} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} n e^{-n} $$