link between infinite products and sums

Let

$$\prod _{k=1}^{\mathrm{\infty }}{p}_k$$

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

$$\sum _{k=1}^{\mathrm{\infty }}\log p_k$$

converges. Moreover

$$\prod _{k=1}^{\mathrm{\infty }}{p}_k=\exp \sum _{k=1}^{\mathrm{\infty }}\log p_k.$$

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)

$$\underset{N\to \mathrm{\infty }}{lim}\prod _{k=1}^N{p}_k=\underset{N\to \mathrm{\infty }}{lim}\exp \sum _{k=1}^N\log p_k=\exp \sum _{k=1}^{\mathrm{\infty }}\log p_k$$

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