Show that the alternating series $\sum\limits_{k\,=\,0}^{\infty }\dfrac{(-1)^{k}}{(2k) !}=1-\dfrac{1}{2}+\dfrac{1}{24}-\dfrac{1}{720}+\cdots$ converges.

Solution We begin by confirming that $\lim\limits_{n\rightarrow \infty }a_{n}=\lim\limits_{n\rightarrow \infty }\dfrac{1}{(2n)!}=0$. Next using the Algebraic Ratio test, we verify that the terms $a_{k}=\dfrac{1}{(2k) !}$ are nonincreasing. Since $$\begin{eqnarray*} \dfrac{a_{n+1}}{a_{n}} &=& \dfrac{\dfrac{1}{[ 2(n+1) ] !} }{\dfrac{1}{(2n) !}}=\dfrac{(2n) !}{( 2n+2) !} =\dfrac{(2n) !}{( 2n+2) (2n+1) (2n) !}\\ &=&\dfrac{1}{( 2n+2) (2n+1) }<1 \qquad \hbox{for all }n\geq 1 \end{eqnarray*}$$

the terms $a_{k}$ are nonincreasing. By the Alternating Series Test, the series converges.