91Ó°ÊÓ

Using sigma notation, write the following expressions as infinite series. $$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $$

Using sigma notation, write the following expressions as infinite series. $$ 1-1+1-1+\cdots $$

Using sigma notation, write the following expressions as infinite series. $$ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots $$

Using sigma notation, write the following expressions as infinite series. $$ \sin 1+\sin 1 / 2+\sin 1 / 3+\sin 1 / 4+\cdots $$

Compute the first four partial sums \(S_{1}, \ldots, S_{4}\) for the series having \(n\) th term \(a_{n}\) starting with \(n=1\) as follows. $$ a_{n}=n $$

Compute the first four partial sums \(S_{1}, \ldots, S_{4}\) for the series having \(n\) th term \(a_{n}\) starting with \(n=1\) as follows. $$ a_{n}=1 / n $$

Compute the first four partial sums \(S_{1}, \ldots, S_{4}\) for the series having \(n\) th term \(a_{n}\) starting with \(n=1\) as follows. $$ a_{n}=\sin (n \pi / 2) $$

Compute the first four partial sums \(S_{1}, \ldots, S_{4}\) for the series having \(n\) th term \(a_{n}\) starting with \(n=1\) as follows. $$ a_{n}=(-1)^{n} $$

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 \(S\). $$ S_{n}=1-\frac{1}{n}, \quad n \geq 2 $$

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 \(S\). $$ S_{n}=\frac{n(n+1)}{2}, \quad n \geq 1 $$