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:

$s_n = \map \log {p_n}$

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Also see

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