Infinite product

In mathematics, for a sequence of complex numbers a₁, a₂, a₃, ... the infinite product

is defined to be the limit of the partial products a₁a₂...a_n as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence a_n as n increases without bound must be 1, while the converse is in general not true.

The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product):

The product of positive real numbers

converges to a nonzero real number if and only if the sum

converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products. The same criterion applies to products of arbitrary complex numbers (including negative reals) if the logarithm is understood as a fixed branch of logarithm which satisfies ln(1) = 0, with the proviso that the infinite product diverges when infinitely many a_n fall outside the domain of ln, whereas finitely many such a_n can be ignored in the sum.

For products of reals in which each ${\displaystyle a_{n}\geq 1}$, written as, for instance, ${\displaystyle a_{n}=1+p_{n},}$ where ${\displaystyle p_{n}\geq 0}$, the bounds

...
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