91Ó°ÊÓ

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. \(f(x)=\sin 2 x, \quad c=0\)

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

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. \(\sum_{n=2}^{\infty}\left(\frac{1}{\ln n}-\frac{1}{\ln (n+1)}\right)\)

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

Find the radius of convergence and the interval of convergence of the power series. $$ \sum_{n=0}^{\infty} \frac{(2 x)^{n}}{n !} $$

Use the Integral Test to determine whether the series is convergent or divergent. $$ \frac{1}{2}+\frac{1}{5}+\frac{1}{10}+\frac{1}{17}+\frac{1}{26}+\cdots $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\) whose \(n\) th term is given. \(a_{n}=\frac{2^{n}}{(2 n) !}\)

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

Use the Comparison Test to determine whether the series is convergent or divergent. \(\sum_{n=0}^{\infty} \frac{1}{\sqrt{n^{3}+1}}\)

Use the Integral Test to determine whether the series is convergent or divergent. $$ \frac{1}{3}+\frac{1}{7}+\frac{1}{11}+\frac{1}{15}+\frac{1}{19}+\cdots $$