91Ó°ÊÓ

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

Determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n 3^{n}}$$

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots+(-1)^{n-1} \frac{1}{2^{n-1}}+\cdots$$

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

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 \(a_{4}\). $$a_{n}=\frac{(-1)^{n+1}}{2 n-1}$$

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely,(c) conditionally? $$\sum_{n=0}^{\infty}(-1)^{n}(4 x+1)^{n}$$

Find the first four nonzero terms of the Taylor series for the functions. $$(1-x)^{-3}$$

Find a formula for the \(n\) th partial sum of each series and use it to find the series' sum if the series converges. $$1-2+4-8+\dots+(-1)^{n-1} 2^{n-1}+\dots$$

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 \(a_{4}\). $$a_{n}=2+(-1)^{n}$$

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. $$\sum_{n=1}^{\infty} \frac{1}{n+4}$$