For questions on infinite products: convergence, computation, etc...

In mathematics, for a sequence of complex numbers $a_1, a_2, a_3, ...$ the infinite product $$ \prod_{n=1}^{\infty} a_n = a_1 \times a_2 \times a_3 \cdots $$ is defined to be the limit of the partial products $a_1a_2... a_n$ as $n$ increases without bound. The product is said to converge when the limit exists and is a non-zero number. If the limit is zero or does not approach a complex number the product is said to diverge: this is to preserve the analogy with positive terms where if both sides converge we have

$$ \prod_{n=1}^{\infty} a_n = \exp\left(\sum_{n=1}^{\infty} \log(a_n)\right) $$

Euler products are a special case where the product is indexed over prime numbers; see euler-product for more information.

Not to be confused with direct-product, product-space, etc.