Absolute Value of Infinite Product

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Theorem

Let $\struct {\mathbb K, \norm{\,\cdot\,}}$ be a valued field.

Let $\sequence{a_n}$ be a sequence in $\mathbb K$.

Convergence

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge to $a\in\mathbb K$.


Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm{a_n}$ converges to $\norm{a}$.


Absolute Convergence

Let the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converge absolutely to $a\in\mathbb K$.


Then $\displaystyle \prod_{n \mathop = 1}^\infty \norm{a_n}$ converges absolutely to $\norm{a}$.


Divergence to zero

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$ if and only if $\displaystyle \prod_{n \mathop = 1}^\infty \norm{a_n}$ diverges to $0$.


Also see