Infinite product
From Conservapedia
An infinite product
of a sequence of terms a1, a2, a3, ... is defined to be the limit of the partial products a1a2...an as n goes to infinity. The infinite product converges if and only if the the infinite sum 
converge.
Infinite Product representation of entire functions
Karl Weierstrass proved that every entire function f(z) with a sequence (λn) of zeros that does not accumulate, can be factored into an infinite product of the form:
![f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{\lambda_n} \right) \;
e^{\left [ \frac{z}{\lambda_n} + \frac12\left(\frac{z}{\lambda_n}\right)^2 + \cdots + \frac1{m_n}\left(\frac{z}{\lambda_n}\right)^{m_n} \right ]}](images/math/9/6/9/969108c542ddb91457f6a0c07a6a467d.png)
where m is the multiplicity of the zero of f(z) at the origin, and φ(z) is some entire function.
One spectacular result of the Weierstrass Factorization Theorem is the representation of the Riemann Zeta function ζ as a product over its non-trivial zeros n, known as the Hadamard Product:
![\zeta(z) = \frac{e^{\left [ ln 2 \pi - 1 - \frac{\gamma}{2} \right ]z}}{2(z-1) \Gamma (1+\frac{z}{2})} \prod_{n} (1-\frac{z}{n})e^{\frac{z}{n}}](images/math/f/d/2/fd20da8bc694fbdf530cbb0463b360b8.png)
where γ is the Euler-Mascheroni constant and Γ is the Gamma function[1].
