Show that the series $\sum\limits_{k\,=\,1}^{\infty }(-1)^{k}=-1+1-1+\cdots$ diverges.

Solution The sequence $\{S_{n}\}$ of partial sums for this series is $$\begin{array}{c@{}l} S_{1} &= -1 \\ S_{2} &= -1+1=0 \\ S_{3} &= -1+1-1=-1 \\ S_{4} &= -1+1-1+1=0 \\ \vdots & \\ S_{n} &= \left\{ \begin{array}{r@{\qquad}l@{\quad}l} -1 & \hbox{if} & n \hbox{ is odd} \\ 0  & \hbox{if} & n \hbox{ is even} \end{array} \right.\end{array}$$

Since $\lim\limits_{n\rightarrow \infty }S_{n}$ does not exist, the sequence $\{S_{n}\}$ of partial sums diverges. Therefore, the series diverges.