link between infinite products and sums


Let

k=1pk

be an infinite product such that pk>0 for all k. Then the infinite product converges if and only if the infinite sum

k=1logpk

converges. Moreover

k=1pk=expk=1logpk.

Proof.

Simply notice that

k=1Npk=expk=1Nlogpk.

If the infinite sum converges then (by continuity of exp function)

limNk=1Npk=limNexpk=1Nlogpk=expk=1logpk

and also the infinite product converges.

Title link between infinite products and sums
Canonical name LinkBetweenInfiniteProductsAndSums
Date of creation 2013-03-22 13:41:38
Last modified on 2013-03-22 13:41:38
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 7
Author paolini (1187)
Entry type Theorem
Classification msc 26E99
Classification msc 40A20