91Ó°ÊÓ

If the series is positive-term, determine whether it is convergent or divergent; if the series contains negative terms, determine whether it is absolutely convergent, conditionally convergent, or divergent. $$ \sum_{n=1}^{\infty} \frac{\tan ^{-1} n}{\sqrt{1+n^{2}}} $$

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{(2 n)^{n}}{\left(5 n+3 n^{-1}\right)^{n}}\)

Use the integral test (11.23) to determine the convergence or divergence of the series. $$ \sum_{n=1}^{x} \frac{1}{(3 n+2)^{3}} $$

Use the power series representation for \(\left(1-x^{2}\right)^{-1}\) to find a power series representation for \(2 x\left(1-x^{2}\right)^{-2}\).

Determine whether the sequence converges or diverges, and if it converges, find the limit. $$ \left\\{\frac{n^{2}}{2 n-1}-\frac{n^{2}}{2 n+1}\right\\} $$

Exer. \(33-40:\) Use the \(n\) th-term test (11.17) to determine whether the series diverges or needs further investigation. $$ \sum_{n=1}^{\infty} \frac{3 n}{5 n-1} $$

Use the first two nonzero terms of a Maclaurin series to approximate the number, and estimate the error in the approximation. \(\tan ^{-1} 0.1\)

Determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{\ln n}{(1.01)^{n}}\)

Approximate the sum of each series to three decimal places. $$ \sum_{n=0}^{\infty}(-1)^{n} \frac{1}{n !} $$

Find the radius of convergence of the power series. $$ \sum_{n=0}^{x} \frac{(n+1) !}{10^{n}}(x-5)^{n} $$