Show that the alternating harmonic series $\sum\limits_{k=1}^{\infty }\dfrac{(-1)^{k+1}}{k}=1-\dfrac{1}{2}+\dfrac{1}{3} -\dfrac{1}{4}+\cdots$ converges.

Solution We check the two conditions of the Alternating Series Test. Since $\lim\limits_{n\,\rightarrow \,\infty} a_{n}=\lim\limits_{n\,\rightarrow \,\infty }\dfrac{1}{n}=0,$ the first condition is met. Since $a_{n+1}=\dfrac{1}{n+1}<\dfrac{1}{n}=a_{n}$, the $a_{n}$ are nonincreasing, so the second condition is met. By the Alternating Series Test, the series converges.