91Ó°ÊÓ

Find a formula for an arbitrary Taylor polynomial of \(f\). $$ f(x)=\ln \frac{1+x}{1-x} $$

Use the Comparison Test, the Limit Comparison Test, or the Integral Test to determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{\cos ^{2} n}{n^{3 / 2}} $$

Find the interval of convergence of the given series. $$ \sum_{n=2}^{\infty} \frac{\ln n}{n} x^{n} $$

Evaluate the limit as a number, \(\infty\), or \(-\infty\), $$ \lim _{j \rightarrow \infty} \frac{3^{j}}{2^{j}} $$

Find the limit. $$ \lim _{n \rightarrow \infty} \sqrt[n]{5}\left(\frac{n-1}{n+1}\right) $$

Find the Taylor series of the given function about \(a\). Use the series already obtained in the text or in previous exercises. $$ f(x)=\sin ^{2} x ; a=0\left(\text { Hint }: \sin ^{2} x=\frac{1}{2}(1-\cos 2 x) .\right) $$

Determine whether the series converges or diverges. In some cases you may need to use tests other than the Ratio and Root Tests. $$ \sum_{n=1}^{\infty} \frac{n+5}{n^{3}} $$

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. $$ \sum_{n=1}^{\infty}\left(\frac{1}{4}\right)^{n} $$

Determine whether the series converges or diverges. In some cases you may need to use tests other than the Ratio and Root Tests. $$ \sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2 \cdot 4 \cdot 6 \cdots(2 n)} $$

Find the limit. $$ \lim _{n \rightarrow \infty} \sqrt[n]{3 n} $$