proof of convergence condition of infinite product

proof of theorem of convergence of infinite productFernando Sanz Gamiz


Let pn=i=1nui. We have to study the convergence of the sequence {pn}. The sequence {pn} converges to a not null limit iff {logpn} (log is restricted to its principal branchMathworldPlanetmath) converges to a finite limit. By the Cauchy criterion, this happens iff for every ϵ>0 there exist N such that |logpn+k-logpn|<ϵ for all n>N and all k=1,2,, i.e, iff


as log(z) is an injective function and continuous at z=1 and log(1)=0 this will happen iff for every ϵ>0


for n greater than N and k=1,2,

Title proof of convergence condition of infinite product
Canonical name ProofOfConvergenceConditionOfInfiniteProduct
Date of creation 2013-03-22 17:22:27
Last modified on 2013-03-22 17:22:27
Owner fernsanz (8869)
Last modified by fernsanz (8869)
Numerical id 5
Author fernsanz (8869)
Entry type Proof
Classification msc 30E20