Definition:Convergent Product/Arbitrary Field

Definition
Let $\mathbb K$ be a field with absolute value $\left\vert{\cdot}\right\vert$.

Let $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ be an infinite product.

Let $\left \langle {p_N} \right \rangle$ be the sequence of partial products of $\displaystyle \prod_{n \mathop = 1}^\infty a_n$.

If $p_N \to p\in\mathbb K\setminus\{0\}$ as $N \to \infty$, the product converges to the product $p$, and one writes:
 * $\displaystyle \prod_{n \mathop = 1}^\infty a_n = p$.

A product is said to be convergent it converges to some $p\in \mathbb K\setminus\{0\}$.