analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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An infinite product is a sequence of numbers (usually real or complex $(a_k)_{k\in\mathbb{N}}$ written as $\prod_{k=0}^\infty a_k$. 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.)
A naive definition of convergence, by analogy with the sum of a series, would be that $\underoverset{k=0}{\infty}{\prod} a_k = \underset{N\to\infty}{\lim} \prod_{k=0}^N a_k$ if the latter limit exists. However, this has the flaw that it could happen that this limit exists and yet $\underset{k\to\infty}{\lim} a_k$ might not, whereas we would like to be able to say that if $\prod_{k=0}^\infty a_k$ converges then $\underset{k\to\infty}{\lim} a_k = 1$ (by analogy with the fact that if $\sum_{k=0}^\infty a_k$ converges then $\underset{k\to\infty}{\lim} a_k = 0$). This failure can happen for two reasons:
If some $a_k = 0$, then $\underset{N\to\infty}{\lim} \prod_{k=0}^N a_k = 0$ since the partial products are eventually $0$, regardless of the eventual behavior of the sequence $(a_k)$.
If ${\vert a_k\vert}\le M\lt 1$, then $\underset{N\to\infty}{\lim} \prod_{k=0}^N a_k = 0$, whereas the sequence $(a_k)$ might approach any limit of absolute value $\lt 1$ or have no limit at all.
To avoid these “pathological” situations, we make the following modified definition.
Suppose at most finitely many of the $a_k$ are zero. We say that $\prod_{k=0}^\infty a_k$ converges if
exists and is nonzero. If this is the case, we say that
If the above limit equals 0, one sometimes says that $\prod_{k=0}^\infty a_k$ diverges to 0.
Last revised on December 4, 2014 at 21:34:33. See the history of this page for a list of all contributions to it.