Infinite Product

the product of an infinite number of factors u₁, u₂,..., u_n,. . .—that is an expression of the form

An infinite product in which the factors are numbers is sometimes called an infinite numerical product. An infinite product cannot always be assigned a numerical value. If there exists a limit of the sequence of partial products

P_n = u₁u₂ . . . u_n

which is distinct from zero as n → ∞, then the infinite product is called convergent, and lim p_n = p is its value. We write

Historically, the infinite product was first encountered in connection with problems concerning the calculation of the number π Thus, the 16th-century French mathematician F. Vieta obtained the formula

and the 17th-century English mathematician J. Wallis the formula

The infinite product acquired special importance after the work of L. Euler, who used the infinite product for the representation of functions. An example is the expansion of sin:

The expansion of functions into infinite products is analogous to the expansion of polynomials into linear factors; they are unusual in that they indicate all values of the independent variable for which the function vanishes.

For the convergence of an infinite product, it is necessary and sufficient that u_n = 0 for all numbers n, that u_N^< 0, starting with some number N, and that the series

converges. Thus, the study of the convergence of an infinite product is equivalent to the study of the convergence of this series.