Explore: Infinite Products

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

formalizations of the notions of mathematical induction, infinite series, infinite products, and limits. The concept of actual infinity was introduced

positive real number. One may also consider products of infinitely many factors; these are called infinite products. Notationally, this consists in replacing

analytic functions can be written as infinite products, and these can sometimes be represented as finite products or quotients of the gamma function. We

theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for

10^{-43}} Schmuland has given appealing probabilistic formulations of the infinite product Borwein integrals. For example, consider the random harmonic series

The Wallis product is the infinite product representation of π: π 2 = ∏ n = 1 ∞ 4 n 2 4 n 2 − 1 = ∏ n = 1 ∞ ( 2 n 2 n − 1 ⋅ 2 n 2 n + 1 ) = ( 2 1 ⋅ 2

But in Arithmetica infinitorum (1656), he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or

occur in its Taylor series. The sum of the sinh and cosh series is the infinite series expression of the exponential function. The following series are

{(-1)^{n}x^{2n}}{(2n)!}}.} For applications to special functions, the following infinite product formulae for trigonometric functions are useful: sin ⁡ x = x ∏ n =