91Ó°ÊÓ

In the following exercises, compute the general term \(a_{n}\) ofthe series with the given partial sum \(S_{n}\). If the sequence of partial sums converges, find its limit \(S\). $$ S_{n}=\sqrt{n}, n \geq 2 $$

In the following exercises, compute the general term \(a_{n}\) ofthe series with the given partial sum \(S_{n}\). If the sequence of partial sums converges, find its limit \(S\). $$ S_{n}=2-(n+2) / 2^{n}, n \geq 1 $$

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{n}{n+2}\)

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. \(\left.\sum_{n=1}^{\infty}\left(1-(-1)^{n}\right)\right)\)

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)}\)

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. \(\sum_{n=1}^{\infty} \frac{1}{2 n+1}\)

Suppose that \(\sum_{n=1}^{\infty} a_{n}=1,\) that \(\sum_{n=1}^{\infty} b_{n}=-1,\) that \(a_{1}=2,\) and \(b_{1}=-3 .\) Find the sum of the indicated series. \(\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right)\)

Suppose that \(\sum_{n=1}^{\infty} a_{n}=1,\) that \(\sum_{n=1}^{\infty} b_{n}=-1,\) that \(a_{1}=2,\) and \(b_{1}=-3 .\) Find the sum of the indicated series. \(\sum_{n=1}^{\infty}\left(a_{n}-2 b_{n}\right)\)

Suppose that \(\sum_{n=1}^{\infty} a_{n}=1,\) that \(\sum_{n=1}^{\infty} b_{n}=-1,\) that \(a_{1}=2,\) and \(b_{1}=-3 .\) Find the sum of the indicated series. \(\sum_{n=2}^{\infty}\left(a_{n}-b_{n}\right)\)

Suppose that \(\sum_{n=1}^{\infty} a_{n}=1,\) that \(\sum_{n=1}^{\infty} b_{n}=-1,\) that \(a_{1}=2,\) and \(b_{1}=-3 .\) Find the sum of the indicated series. \(\sum_{n=1}^{\infty}\left(3 a_{n+1}-4 b_{n+1}\right)\)