91Ó°ÊÓ

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{2^{n}}{n^{2}}$$

Find the first four terms of the binomial series for the functions. $$\left(1+x^{3}\right)^{-1 / 2}$$

Use the Comparison Test to determine if each series converges or diverges. $$\sum_{n=1}^{\infty} \sqrt{\frac{n+4}{n^{4}+4}}$$

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a\). $$f(x)=\tan x, \quad a=\pi / 4$$

Use the Ratio Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{n 5^{n}}{(2 n+3) \ln (n+1)}$$

Use the Comparison Test to determine if each series converges or diverges. $$\sum_{n=1}^{\infty} \frac{\sqrt{n}+1}{\sqrt{n^{2}+3}}$$

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n} \frac{10^{n}}{(n+1) !}$$

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

Gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. \(a_{1}=1, \quad a_{n+1}=a_{n} /(n+1)\)

Find the first four terms of the binomial series for the functions. $$\left(1+x^{2}\right)^{-1 / 3}$$