Product of Convergent and Divergent Product is Divergent

Theorem

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

Let $\ds \prod_{n \mathop = 1}^\infty a_n$ be convergent.

Let $\ds \prod_{n \mathop = 1}^\infty b_n$ be divergent.

Then $\ds \prod_{n \mathop = 1}^\infty a_n b_n$ is divergent.

Proof

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Also see

• Product of Convergent Products is Convergent
• Product of Absolutely Convergent Products is Absolutely Convergent