91Ó°ÊÓ

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

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. $$\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}}$$

Find the first four terms of the binomial series for the functions. $$\left(1-\frac{x}{3}\right)^{4}$$

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

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

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=0}^{\infty} \frac{n x^{n}}{n+2}$$

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

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

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. $$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{4^{n}}$$

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