Absolute Value of Infinite Product

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 $\ds \prod_{n \mathop = 1}^\infty a_n$ converge to $a \in \mathbb K$.

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

Absolute Convergence

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

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

Divergence to zero

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

Also see

• Absolute Value of Uniformly Convergent Product