Logarithm of Divergent Product of Real Numbers/Infinity

Theorem
Let $\sequence {a_n}$ be a sequence of strictly positive real numbers.


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

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:
 * $s_n = \map \log {p_n}$

Also see

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