alternating series test

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

Theorem [1, 2] Let ${(a_n)}_{n=1}^{\mathrm{\infty }}$ be a non-negative, non-increasing sequence or 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 $\ln (2)$.

Title	alternating series test
Canonical name	AlternatingSeriesTest
Date of creation	2013-03-22 12:27:09
Last modified on	2013-03-22 12:27:09
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	18
Author	Koro (127)
Entry type	Theorem
Classification	msc 40A05
Classification	msc 40-00
Synonym	Leibniz’s theorem
Synonym	Leibniz test
Related topic	AlternatingSeries