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 $\ds \sum_{n \mathop = 1}^\infty \log a_n$.

By Sum of Logarithms:
 * $s_n = \map \log {p_n}$

Also see

 * Logarithm of Infinite Product of Real Numbers, for similar results