Proof of Stolz-Cesaro theorem


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

l-ϵ<an+1-anbn+1-bn<l+ϵ

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

(l-ϵ)(bn+1-bn)<an+1-an<(l+ϵ)(bn+1-bn)

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

(l-ϵ)i=N(ϵ)k(bi+1-bi)<i=N(ϵ)k(an+1-an)<(l+ϵ)i=N(ϵ)k(bi+1-bi)
(l-ϵ)(bk+1-bN(ϵ))<ak+1-aN(ϵ)<(l+ϵ)(bk+1-bN(ϵ))

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

(l-ϵ)(1-bN(ϵ)bk+1)<ak+1bk+1-aN(ϵ)bk+1<(l+ϵ)(1-bN(ϵ)bk+1)
(l-ϵ)(1-bN(ϵ)bk+1)+aN(ϵ)bk+1<ak+1bk+1<(l+ϵ)(1-bN(ϵ)bk+1)+aN(ϵ)bk+1

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

(l-ϵ)<ak+1bk+1<(l+ϵ)

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

This obviously means that :

limnanbn=l

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
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Last modified by slash (33)
Numerical id 4
Author slash (33)
Entry type Proof
Classification msc 40A05