proof of Leibniz’s theorem (using Dirichlet’s convergence test)


Proof. Let us define the sequenceMathworldPlanetmath αn=(-1)n for n={0,1,2,}. Then

i=0nαi={1for evenn,0for oddn,

so the sequence i=0nαi is bounded. By assumptionPlanetmathPlanetmath {an}n=1 is a bounded decreasing sequence with limit 0. For n we set bn:=an+1. Using Dirichlet’s convergence test, it follows that the series i=0αibi convergesPlanetmathPlanetmath. Since

i=0αibi=n=1(-1)n+1an,

the claim follows.

Title proof of Leibniz’s theorem (using Dirichlet’s convergence test)
Canonical name ProofOfLeibnizsTheoremusingDirichletsConvergenceTest
Date of creation 2013-03-22 13:22:17
Last modified on 2013-03-22 13:22:17
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Proof
Classification msc 40A05
Related topic AlternatingSeries