91Ó°ÊÓ

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. $$\sum_{n=2}^{\infty} \frac{1}{4^{n}}$$

Use the Integral Test to determine whether the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied. $$\sum_{n=1}^{\infty} \frac{n^{2}}{e^{n / 3}}$$

Use the limit Comparison Test to determine whether each series converges or diverges. $$\sum_{n=1}^{\infty} \frac{n-2}{n^{3}-n^{2}+3}$$ (Hint: limit Comparison with \(\left.\sum_{n=1}^{\infty}\left(1 / n^{2}\right)\right)\)

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

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

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

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

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

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

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