91Ó°ÊÓ

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

Determine if 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 the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a\). $$f(x)=\ln x, \quad a=1$$

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

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

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 if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{2^{n+1}}{n^{n-1}}$$

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 if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$\sum_{n=1}^{\infty} \frac{1}{n+4}$$

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