91Ó°ÊÓ

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.\(^{n \rightarrow \infty}\) $$\sum_{k=1}^{\infty}\left(\frac{1}{k+1}-\frac{1}{k+2}\right)$$

A ball is thrown upward 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 nth bounce. Consider the following values of \(h_{0}\) and \(r\). $$h_{0}=20, r=0.5$$

Find the limit of the following sequences or state that they diverge. $$\left\\{\frac{\sin n}{2^{n}}\right\\}$$

A ball is thrown upward 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 nth bounce. Consider the following values of \(h_{0}\) and \(r\). $$h_{0}=10, r=0.9$$

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

Use the test of your choice to determine whether the following series converge. $$\sum_{k=1}^{\infty}\left(1-\frac{1}{k}\right)^{k^{2}}$$

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.\(^{n \rightarrow \infty}\) $$\sum_{k=1}^{\infty}\left(\frac{1}{k+2}-\frac{1}{k+3}\right)$$

Find the limit of the following sequences or state that they diverge. $$\left\\{\frac{\cos (n \pi / 2)}{\sqrt{n}}\right\\}$$

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

A ball is thrown upward 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 nth bounce. Consider the following values of \(h_{0}\) and \(r\). $$h_{0}=30, r=0.25$$