Definition:Convergent Product/Number Field

Definition

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 $\mathbb K$.

Then:

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:

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

Sources

• 1973: John B. Conway: Functions of One Complex Variable ... (next) $VII$: Compact and Convergence in the Space of Analytic Functions: $\S5$: Weierstrass Factorization Theorem: Definition $5.1$