Logarithm of Convergent Product of Real Numbers

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Theorem

Let $\left\langle{a_n}\right\rangle$ 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 $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ converges to $\log a$.


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

By Sum of Logarithms, $s_n = \log \left({p_n}\right)$.


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, $s_n \to \log a$.

$\Box$


2 implies 1

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

Then $s_n \to \log a$.

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

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

$\blacksquare$


Also see