91Ó°ÊÓ

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} \frac{k !}{k^{k}+3}$$

Use the Ratio Test or the Root Test to determine the values of \(x\) for which each series converges. $$\sum_{k=1}^{\infty} x^{k}$$

Heights of bouncing balls A ball is thrown upwand to a height of \(h_{0}\) meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nath bounce. Consider the following values of \(h_{0}\) and \(r\) a. Find the first four terms of the sequence of heights \(\left\\{h_{n}\right\\}\). b. Find an explicit formula for the nith term of the sequence \(\left\\{h_{n}\right\\}\). $$h_{0}=30, r=0.25$$

More sequences Find the limit of the following sequences or determine that the sequence diverges. $$\left\\{\frac{n \sin ^{3}(n \pi / 2)}{n+1}\right\\}$$

Determine whether the following series converge absolutely, converge conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k} \tan ^{-1} k}{k^{3}}$$

Heights of bouncing balls A ball is thrown upwand to a height of \(h_{0}\) meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nath bounce. Consider the following values of \(h_{0}\) and \(r\) a. Find the first four terms of the sequence of heights \(\left\\{h_{n}\right\\}\). b. Find an explicit formula for the nith term of the sequence \(\left\\{h_{n}\right\\}\). $$h_{0}=20, r=0.75$$

Determine whether the following series converge absolutely, converge conditionally, or diverge. $$\sum_{k=1}^{\infty}(-1)^{k} \frac{k^{2}+1}{3 k^{4}+3}$$

Use the Ratio Test or the Root Test to determine the values of \(x\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k}$$

Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums \(\left\\{S_{n}\right\\} .\) Then evaluate lim \(S_{n}\) to obtain the value of the series or state that the series diverges. \(^{n \rightarrow \infty}\). $$\sum_{k=0}^{\infty} \frac{1}{(3 k+1)(3 k+4)}$$

More sequences Find the limit of the following sequences or determine that the sequence diverges. $$\left\\{\frac{(-1)^{n+1} n^{2}}{2 n^{3}+n}\right\\}$$