# Logarithm of Divergent Product of Real Numbers/Zero

Jump to navigation
Jump to search

## 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 $0$.

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

- $s_n = \map \log {p_n}$

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also see

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