91Ó°ÊÓ

In Exercises \(1-10\), 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. \(f(x)=e^{2 x}, \quad c=0\)

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. \(\sum_{n=1}^{\infty}\left(\frac{-8}{4 n^{2}+4 n-3}\right)\)

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

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

Find an expression for the \(n\) th term of the sequence. (Assume that the pattern continues.) \(\\{0,2,0,2,0, \ldots\\}\)

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

Find all values of \(x\) for which the series \(\sum_{n=1}^{\infty} \frac{x^{n}}{n}\) (a) converges absolutely and (b) converges conditionally.

In Exercises \(35-40\), find the first three terms of the Taylor series of \(f\) at the given value of \(c\). \(f(x)=\tan x, \quad c=\frac{\pi}{4}\)

Use the Maclaurin series for \(e^{-x^{2}}\) to calculate \(e^{-0.01}\) accurate to five decimal places.

Use the Maclaurin series for \(\cos x\) to calculate \(\cos 3^{\circ}\) accurate to five decimal places.