Definition:Convergent Product/Number Field

Definition
Let $\mathbb K$ be one of the standard number fields $\Q, \R, \C$.

Let $(a_n)$ be a sequence of elements of $\mathbb K$.

Then the infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is convergent :
 * There exists $n_0\in\N$ such that $a_n\neq0$ for $n> n_0$.
 * The sequence of partial products of $\displaystyle \prod_{n \mathop = n_0+1}^\infty a_n$ converges to some $b\in\mathbb K\setminus\{0\}$.

The product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is said to be convergent to $a=a_1\cdots a_{n_0}\cdot b$, and one writes:
 * $\displaystyle \prod_{n \mathop = 1}^\infty a_n = a$

A product is thus convergent it converges to some $a\in \mathbb K$.

Also see

 * Value of Convergent Infinite Product is Well-Defined, for a proof that the assigned value does not depend on the choice of $n_0$