91Ó°ÊÓ

Each 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. $$a_{1}=1, \quad a_{n+1}=a_{n}+\left(1 / 2^{n}\right)$$

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

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

Use substitution to find the Taylor series at \(x=0\) of the functions. $$\ln \left(1+x^{2}\right)$$

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

Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{4^{n}}$$

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

Each 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. $$a_{1}=1, \quad a_{n+1}=a_{n} /(n+1)$$

Write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. $$\sum_{n=2}^{\infty} \frac{1}{4^{n}}$$