Proof of Stolz-Cesaro theorem

From the definition of convergence , for every ϵ>0 there is N(ϵ) such that ()nN(ϵ) , we have :


Because bn is strictly increasingPlanetmathPlanetmath we can multiply the last equation with bn+1-bn to get :


Let k>N(ϵ) be a natural numberMathworldPlanetmath . Summing the last relationMathworldPlanetmath we get :


Divide the last relation by bk+1>0 to get :


This means that there is some K such that for kK we have :


(since the other terms who were left out convergePlanetmathPlanetmath to 0)

This obviously means that :


and we are done .

Title Proof of Stolz-Cesaro theorem
Canonical name ProofOfStolzCesaroTheorem
Date of creation 2013-03-22 13:17:45
Last modified on 2013-03-22 13:17:45
Owner slash (33)
Last modified by slash (33)
Numerical id 4
Author slash (33)
Entry type Proof
Classification msc 40A05