91Ó°ÊÓ

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

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=1}^{\infty} \frac{(-1)^{n}(x+2)^{n}}{n}$$

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

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=1}^{\infty} \frac{x^{n}}{n \sqrt{n} 3^{n}}$$

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

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. \(a_{1}=2, \quad a_{n+1}=(-1)^{n+1} a_{n} / 2\)

Use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{7}{(2 n+5)^{n}}$$

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

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. $$\sum_{n=1}^{\infty}\left(1-\frac{7}{4^{n}}\right)$$

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. $$\sum_{n=1}^{\infty} \frac{n^{2}}{e^{n / 3}}$$