# Definition:Convergent Product

## Definition

### Convergent Product in a Standard Number Field

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

Then:

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

### Convergent Product in Arbitrary Field

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 $a \in \mathbb K \setminus \set 0$.

### Convergent Product in Normed Algebra

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:

$(1): \quad a_n$ is invertible for $n \ge n_0$
$(2): \quad$ the sequence of partial products of $\ds \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $b\in A^\times$.

### Divergent Product

An infinite product which is not convergent is divergent.

### 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:

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

## Also see

• Results about convergent products can be found here.