91Ó°ÊÓ

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty}\left(\frac{k}{k+10}\right)^{-k^{2}}$$

Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. $$a_{n+1}=\sqrt{8 a_{n}+9} ; a_{1}=10$$

Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. $$\sum_{k=1}^{\infty} \frac{1}{k^{2}+7 k+12}$$

Choose your test Use the test of your choice to determine whether the following series converge. $$\sum_{k=2}^{\infty} \frac{1}{k^{\ln k}}$$

Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. $$\sum_{k=1}^{\infty}\left(\frac{5}{6}\right)^{-k}$$

More sequences Find the limit of the following sequences or determine that the sequence diverges. $$a_{n}=(-1)^{n} \sqrt[n]{n}$$

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 \(\left\\{h_{n}\right\\}\). $$h_{0}=20, r=0.5$$

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

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. \(^{n \rightarrow \infty}\). $$\sum_{k=1}^{\infty} \frac{1}{(k+6)(k+7)}$$

Determine whether the following series converge. Justify your answers. $$\sum_{k=1}^{\infty} 5^{1-2 k}$$