91Ó°ÊÓ

Use the Integral Test to determine whether the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$\sum_{n=1}^{\infty} \frac{1}{n^{2}}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely,(c) conditionally? $$\sum_{n=0}^{\infty} x^{n}$$

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+\dots+\frac{2}{3^{n-1}}+\dots$$

Find the first four nonzero terms of the Taylor series for the functions. $$(1+x)^{1 / 2}$$

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a\) $$f(x)=e^{2 x}, \quad a=0$$

Determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\sqrt{n}}$$

Each of Exercises gives a formula for the \(n\) th term \(a_{n}\) of a sequence \(\left\\{a_{n}\right\\} .\) Find the values of \(a_{1}, a_{2}, a_{3},\) and \(a_{4}\). $$a_{n}=\frac{1-n}{n^{2}}$$

Use the Direct Comparison Test to determine whether each series converges or diverges. $$\sum_{n=1}^{\infty} \frac{1}{n^{2}+30}$$

Use the Ratio Test to determine whether each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$$

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$\frac{9}{100}+\frac{9}{100^{2}}+\frac{9}{100^{3}}+\dots+\frac{9}{100^{n}}+\cdots$$