nLab
infinite product

Infinite products

Infinite products

Definition

An infinite product is a sequence of numbers (usually real or complex (a k) k(a_k)_{k\in\mathbb{N}} written as k=0 a k\prod_{k=0}^\infty a_k. Like an infinite series, we are interested in knowing whether such a product converges, and if so, what it converges to.

(Another sort of infinite product is the cartesian product of an infinite family of sets, or more generally objects of some category. For that notion, see product.)

Convergence

A naive definition of convergence, by analogy with the sum of a series, would be that k=0a k=limN k=0 Na k\underoverset{k=0}{\infty}{\prod} a_k = \underset{N\to\infty}{\lim} \prod_{k=0}^N a_k if the latter limit exists. However, this has the flaw that it could happen that this limit exists and yet limka k\underset{k\to\infty}{\lim} a_k might not, whereas we would like to be able to say that if k=0 a k\prod_{k=0}^\infty a_k converges then limka k=1\underset{k\to\infty}{\lim} a_k = 1 (by analogy with the fact that if k=0 a k\sum_{k=0}^\infty a_k converges then limka k=0\underset{k\to\infty}{\lim} a_k = 0). This failure can happen for two reasons:

  1. If some a k=0a_k = 0, then limN k=0 Na k=0\underset{N\to\infty}{\lim} \prod_{k=0}^N a_k = 0 since the partial products are eventually 00, regardless of the eventual behavior of the sequence (a k)(a_k).

  2. If |a k|M<1{\vert a_k\vert}\le M\lt 1, then limN k=0 Na k=0\underset{N\to\infty}{\lim} \prod_{k=0}^N a_k = 0, whereas the sequence (a k)(a_k) might approach any limit of absolute value <1\lt 1 or have no limit at all.

To avoid these “pathological” situations, we make the following modified definition.

Definition

Suppose at most finitely many of the a ka_k are zero. We say that k=0 a k\prod_{k=0}^\infty a_k converges if

limN k=1a k0 a k \underset{N\to\infty}{\lim} \prod_{k=1\quad a_k\neq 0}^\infty a_k

exists and is nonzero. If this is the case, we say that

k=0 a k={0 if some a k=0 the above limit otherwise \prod_{k=0}^\infty a_k = \begin{cases} 0 &\quad \text{if some }a_k = 0\\ \text{the above limit} &\quad\text{otherwise} \end{cases}

If the above limit equals 0, one sometimes says that k=0 a k\prod_{k=0}^\infty a_k diverges to 0.

Examples

Last revised on December 4, 2014 at 21:34:33. See the history of this page for a list of all contributions to it.