91Ó°ÊÓ

Determine whether the series converges or diverges. In some cases you may need to use tests other than the Ratio and Root Tests. $$ \sum_{n=1}^{\infty} \frac{\sin 1 / n !}{\cos 1 / n !} $$

Use the Comparison Test, the Limit Comparison Test, or the Integral Test to determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{\tan ^{-1} n}{n^{2}+1} $$

Find the \(n\) th Taylor polynomial of \(f\) for the given values of \(n\). $$ f(x)=\ln (\cos x) ; n=2 $$

Find the limit. $$ \lim _{n \rightarrow \infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n}) $$

Find a formula for the partial sums of the series. For each series, determine whether the partial sums have a limit. If so, find the sum of the series. $$ \sum_{n=1}^{\infty}\left[n^{3}-(n+1)^{3}\right] $$

Find the Taylor series of the given function about \(a\). Use the series already obtained in the text or in previous exercises. $$ f(x)=\sin x ; a=\pi / 3 $$

Find the radius of convergence of the given series. $$ \sum_{n=1}^{\infty} \frac{n !}{n^{n}} x^{n} $$

Approximate the sum of the given series with an error less than \(0.001\). $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n^{3}} $$

Evaluate the limit as a number, \(\infty\), or \(-\infty\), $$ \lim _{n \rightarrow \infty} \frac{2 n^{2}-4}{-n-5} $$

Approximate the sum of the given series with an error less than \(0.001\). $$ \sum_{n=2}^{\infty}(-1)^{n+1} \frac{1}{1+n+6 n^{2}} $$