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of a [[sequence]] of numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... is defined to be the [[limit (mathematics)|limit]] of the partial products ''a''<sub>1</sub>''a''<sub>2</sub>...''a''<sub>''n''</sub> as ''n'' goes to infinity. The infinite product is said to [[convergence|converge]] when the limit exists and is not zero. Otherwise the product is said to [[diverge]].
of a [[sequence]] of numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... is defined to be the [[limit (mathematics)|limit]] of the partial products ''a''<sub>1</sub>''a''<sub>2</sub>...''a''<sub>''n''</sub> as ''n'' goes to infinity. The infinite product is said to [[convergence|converge]] when the limit exists and is not zero. Otherwise the product is said to [[diverge]].
==Infinite Product representations of entire functions==
[[Karl Weierstrass]] proved that every [[entire function]] ''f''(''z'') with a divergent sequence (λ<sub>''n''</sub>) of zeros, can be factored into an infinite product of the form
:<math>
f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{\lambda_n} \right) \;
\exp \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 ]
</math>
where ''m'' is the multiplicity of the zero of ''f''(''z'') at the origin, and φ(''z'') is some [[entire function]].
[[category:mathematics]]
[[category:mathematics]]