Logarithm of Divergent Product of Real Numbers

Theorem
Let $(a_n)$ be a sequence of strictly positive real numbers.

Then the following are equivalent:
 * The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$.


 * The series $\displaystyle \sum_{n \mathop = 1}^\infty\log a_n$ diverges to $-\infty$.

and the following are equivalent:


 * The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$.


 * The series $\displaystyle \sum_{n \mathop = 1}^\infty\log a_n$ diverges to $+\infty$.

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

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

By Sum of Logarithms, $s_n = \log(p_n)$.

Also see

 * Logarithm of Convergent Product of Real Numbers
 * Logarithm of Infinite Product of Complex Numbers