Logarithm of Divergent Product of Real Numbers

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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::

$(1): \quad$ The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$.
$(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $+\infty$.


Also see