# Definition:Convergent Product

## Contents

## Convergent Product in a Standard Number Field

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

### Nonzero Sequence

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

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

### Arbitrary Sequence

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

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

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

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

## Convergent Product in Arbitrary Field

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 $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ 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 $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is **convergent** if and only if:

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

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

## Convergent Product in Normed Algebra

Let $\mathbb K$ be a division ring with norm $\norm{\,\cdot\,}_{\mathbb K}$.

Let $\struct{ A, \norm{\,\cdot\,} }$ be an associative normed unital algebra over $\mathbb K$.

Let $\sequence{a_n}$ be a sequence in $A$.

### Definition 1

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is **convergent** if and only if there exists $n_0\in\N$ such that:

- $a_n$ is invertible for $n \geq n_0$
- the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $b\in A^\times$.

### Definition 2: for complete algebras

Let $\struct{ A, \norm{\,\cdot\,} }$ be complete.

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ is **convergent** if and only if there exists $n_0\in\N$ such that:

- the sequence of partial products of $\displaystyle \prod_{n \mathop = n_0}^\infty a_n$ converges to some invertible $a\in A^\times$.

## 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 $\displaystyle \prod_{n \mathop = n_0 + 1}^\infty a_n$ converges to $0$

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

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

## Also defined as

Some authors define an infinite product to be convergent if and only if the sequence of partial products converges. The phrase:

- $\displaystyle \prod_{n \mathop = 1}^\infty a_n$
**diverges to $0$**

then becomes:

- $\displaystyle \prod_{n \mathop = 1}^\infty a_n$
**converges to $0$**.

The definition used here has many desirable consequences, such as:

- Factors in Convergent Product Converge to One
- Convergence of Infinite Product Does not Depend on Finite Number of Factors
- Product of Convergent and Divergent Product is Divergent

which creates an analogy with series.