# Logarithm of Divergent Product of Real Numbers

## Theorem

Let $\left\langle{a_n}\right\rangle$ be a sequence of strictly positive real numbers.

### Divergence to zero

The following are equivalent:

• The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $-\infty$.

### Divergence to infinity

The following are equivalent:

• The series $\displaystyle \sum_{n \mathop = 1}^\infty\log a_n$ diverges to $+\infty$.