# Logarithm of Divergent Product of Real Numbers/Zero

## Theorem

Let $\sequence {a_n}$ be a sequence of strictly positive real numbers.

The following are equivalent:

The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$.
The series $\ds \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $-\infty$.

## Proof

Let $p_n$ denote the $n$th partial product of $\ds \prod_{n \mathop = 1}^\infty a_n$.

Let $s_n$ denote the $n$th partial sum of $\ds \sum_{n \mathop = 1}^\infty \log a_n$.

$s_n = \map \log {p_n}$