Logarithm of Convergent Product of Real Numbers

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

The following are equivalent:
 * The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$.


 * The series $\displaystyle \sum_{n \mathop = 1}^\infty \ln a_n$ converges to $\ln a$.

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 $\displaystyle \sum_{n \mathop = 1}^\infty \ln a_n$.

By Sum of Logarithms, $s_n = \map \ln {p_n}$.

1 implies 2
Let $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge to $a>0$.

Then $p_n \to a$.

By Natural Logarithm Function is Continuous, $s_n \to \ln a$.

2 implies 1
Let $\displaystyle \sum_{n \mathop = 1}^\infty \ln a_n$ converge to $\ln a$.

Then $s_n \to \ln a$.

By Exponential Function is Continuous, $p_n \to a$.

Because $a \ne 0$, $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $a$.

Also see

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