# link between infinite products and sums

Let

 $\prod_{k=1}^{\infty}p_{k}$

be an infinite product such that $p_{k}>0$ for all $k$. Then the infinite product converges if and only if the infinite sum

 $\sum_{k=1}^{\infty}\log p_{k}$

converges. Moreover

 $\prod_{k=1}^{\infty}p_{k}=\exp\sum_{k=1}^{\infty}\log p_{k}.$

Proof.

Simply notice that

 $\prod_{k=1}^{N}p_{k}=\exp\sum_{k=1}^{N}\log p_{k}.$

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

 $\lim_{N\to\infty}\prod_{k=1}^{N}p_{k}=\lim_{N\to\infty}\exp\sum_{k=1}^{N}\log p% _{k}=\exp\sum_{k=1}^{\infty}\log p_{k}$

and also the infinite product converges.

Title link between infinite products and sums LinkBetweenInfiniteProductsAndSums 2013-03-22 13:41:38 2013-03-22 13:41:38 paolini (1187) paolini (1187) 7 paolini (1187) Theorem msc 26E99 msc 40A20