alternating series test

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

Theorem [1, 2] Let (an)n=1 be a non-negative, non-increasing sequence or real numbers such that limnan=0. Then the infinite series n=1(-1)(n+1)an converges.

This test provides a necessary and sufficient condition for the convergence of an alternating seriesMathworldPlanetmath, since if n=1an converges then an0.

Example: The series k=11k does not converge, but the alternating series k=1(-1)k+11k converges to ln(2).


  • 1 W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Inc., 1976.
  • 2 E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.
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