# Definition:Convergent Product/Arbitrary Field

## Definition

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

### Nonzero Sequence

Let $\sequence {a_n}$ be a sequence of nonzero elements of $\mathbb K$.

The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ is **convergent** if and only if its sequence of partial products converges to a nonzero limit $a \in \mathbb K \setminus \set 0$.

### Arbitrary Sequence

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

The infinite product $\ds \prod_{n \mathop = 1}^\infty a_n$ is **convergent** if and only if:

- there exists $n_0 \in \N$ such that the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty a_n$ converges to some $b \in \mathbb K \setminus \set 0$.

The sequence of partial products of $\ds \prod_{n \mathop = 1}^\infty a_n$ is then convergent to some $a \in \mathbb K$.

The product is said to be **convergent** to $a$, and one writes:

- $\ds \prod_{n \mathop = 1}^\infty a_n = a$

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

## Divergent Product

An infinite product which is not convergent is **divergent**.

This article, or a section of it, needs explaining.In particular: Nelson separately defines an Definition:Oscillating Product which is one that is neither convergent nor divergent, but then does not rigorously define divergent. Research needed.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

### Divergence to zero

If either:

- there exist infinitely many $n \in \N$ with $a_n = 0$

- there exists $n_0 \in \N$ with $a_n \ne 0$ for all $n > n_0$ and the sequence of partial products of $\ds \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$

the product **diverges to $0$**, and we assign the value:

- $\ds \prod_{n \mathop = 1}^\infty a_n = 0$