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An expression of the form
$$
\prod_{k=1}^\infty \left({ 1 + u_k }\right)
\label{(*)}
$$
containing an infinite set of factors, numbers or functions, all of which are non-zero. An infinite product is said to be ''convergent'' if there exists a non-zero limit of the sequence of partial products
$$
P_n = \prod_{k=1}^n \left({ 1 + u_k }\right)
$$
as $n \rightarrow \infty$. The value of the infinite product is the limit
$$
P = \lim_{n\rightarrow\infty} P_n
$$
and one writes
$$
\prod_{k=1}^\infty \left({ 1 + u_k }\right) = P \ .
$$

An infinite product converges if and only if the series
$$
\sum_{k=1}^\infty \log \left({ 1 + u_k }\right)
$$
is convergent. Accordingly, the study of the convergence of infinite products is reduced to the study of the convergence of series. The infinite product (*) is said to be absolutely convergent if the infinite product
$$
\prod_{k=1}^\infty \left({ 1 + |u_k| }\right)
$$
is convergent. A necessary and sufficient condition for absolute convergence of the infinite product (*) is absolute convergence of the series
$$
\sum_{k=1}^\infty  u_k   \ .
$$

An infinite product has the rearrangement property (i.e. its value is independent of the order of the factors) if and only if it is absolutely convergent.

The infinite product (*) with factors which are functions
$$
\left({ 1 + u_k }\right) = \left({ 1 + u_k(z) }\right)
$$
defined, for example, in a domain $D$ of the complex $z$-plane, converges uniformly in $D$ if the sequence of partial products $P_n(z)$ converges uniformly in $D$ to a non-zero limit. A very important case in practical applications is when certain factors have zeros in $D$ such that at most a finite number of the zeros lie in any compact set $K \subseteq D$. The concept of convergence is generalized as follows: The infinite product (*) is said to be (absolutely, uniformly) convergent inside $D$ if for any compact set $K \subseteq D$ there exists a number $N = N(K)$ such that all the factors $\left({ 1 + u_k(z) }\right) \neq 0$ for $k \ge N$, while the sequence of partial products
$$
\prod_{k=N}^n \left({ 1 + u_k(z) }\right)
$$
converges (absolutely, uniformly) on $K$ to a non-zero limit. If all factors are analytic functions in $D$ and if the infinite product converges uniformly inside $D$, its limit is an analytic function in $D$.

Infinite products were first encountered by F. Viète (1593) in his study of the quadrature of the circle. He represented the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050880/i05088027.png" /> analytically by the following infinite product:
$$
\frac{2}{\pi} = \sqrt{ \frac{1}{2} } \cdot \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} } } \cdot \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} } } } \cdots \ .
$$

Another representation of $\pi$ is due to J. Wallis (1665):
$$
\frac{4}{\pi} = \frac32 \cdot \frac34 \cdot \frac54 \cdot \frac56 \cdot \frac 76 \cdot \frac78 \cdots \ .
$$

Infinite products with factors that are functions were encountered by L. Euler (1742); an example is
$$
\sin z = z \prod_{k=1}^\infty \left({ 1 - \frac{z^2}{k^2\pi^2} }\right) \ .
$$

Infinite products are a principal tool in representing analytic functions with explicit indication of their zeros; for [[entire function]]s they are the analogue of the factors of polynomials. See also [[Blaschke product]]; [[Weierstrass theorem]] on infinite products; [[Canonical product]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1''' , MIR  (1982)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  A.V. Bitsadze,   "Fundamentals of the theory of analytic functions of a complex variable" , Moscow  (1969)  (In Russian) {{ZBL|0183.33601}}</TD></TR>
</table>



====Comments====
See also [[Hadamard theorem]] on entire functions.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,   "Functions of one complex variable" , Springer  (1984) {{ZBL|0277.30001}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.S.B. Holland,   "Introduction to the theory of entire functions" , Acad. Press  (1973) {{ZBL|0278.30001}}</TD></TR>
</table>

====Comments====
It should be noted in the definition above that an infinite product is said to converge if the limit of the sequence of partial products $P_n$ is non-zero.  If the limit is zero than the product is said to ''diverge to zero''.  

====References====
<table>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  R.A. Rankin, "An Introduction to Mathematical Analysis", Pergamon Press (1963) {{ZBL|0112.28103}}</TD></TR>
</table>

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