Logarithm of Convergent Product of Real Numbers

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


 * $(1): \quad$ The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$.
 * $(2): \quad$ The series $\ds \sum_{n \mathop = 1}^\infty \ln a_n$ converges to $\ln a$.

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

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

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

Then $p_n \to a$.

By Real Natural Logarithm Function is Continuous:
 * $s_n \to \ln a$

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

Then $s_n \to \ln a$.

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

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

Also see

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