alternating series test

The alternating series test, orthe Leibniz’s Theorem, states the following:

Theorem [1, 2]Let ${(a_n)}_{n=1}^{\mathrm{\infty }}$ be a non-negative, non-increasing sequenceor real numbers such that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.Then the infinite series ${\sum }_{n=1}^{\mathrm{\infty }}{(-1)}^{(n+1)}{a}_n$ converges.

This test provides a necessary and sufficient condition for the convergence of an alternating series, since if ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ converges then $a_n\to 0$.

Example: The series${\sum }_{k=1}^{\mathrm{\infty }}\frac{1}{k}$does not converge, but the alternating series${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{1}{k}$converges to $\ln (2)$.