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.