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 an 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
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 an fall outside the domain of ln, whereas finitely many such an can be ignored in the sum.
and by the limit comparison test it follows that the two series
are equivalent meaning that either they both converge or they both diverge.
Product representations of functions
Main article: Weierstrass factorization theorem
One important result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions, each with at most a single root.
In general, if f has a root of order m at the origin and has other complex roots at u1, u2, u3, ... (listed with multiplicities equal to their orders), then
This can be regarded as a generalization of the fundamental theorem of algebra, since for polynomials, the product becomes finite and φ(z) is constant.
In addition to these examples, the following representations are of special note:
The last of these is not a product representation of the same sort discussed above, as ζ is not entire.
Rather, the above product representation of ζ(z) converges precisely for Re(z) > 1, where it is an analytic function.
By techniques of analytic continuation, this function can be extended uniquely to an analytic function (still denoted ζ(z)) on the whole complex plane except at the point z = 1, where it has a simple pole.
- Infinite products in trigonometry
- Infinite series
- Continued fraction
- Infinite expression
- Iterated binary operation
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Infinite product.