Infinite Products, The Infinite Product of Infinite Series

The Infinite Product of Infinite Series

Let M be a matrix of nonnegative real numbers. Each row of M represents a (possibly different) convergent series. Let t_i be the sum of the i^th series, defined by the i^th row of M. Can we interchange product and summation, as we did in the previous theorem? Is there a sum of products that is equal to the product of these sums? There is, if each series includes the number 1. Thus each t_i ≥ 1.

Let the product over t be q. Thus the partial products of t approach q from below, increasing monotonically.

Let w_n be the sum of products, where each product selects one term from each of the first n rows of M. Thus w₁ is the sum of the first row, or t₁. Next, w₂ is the sum of M_1,iM_2,j for i and j > 0. By the previous theorem, this is the same as t₁×t₂. Since the second row of M includes 1, a copy of w₁ is present in w₂. In general, w_n is equal to the product t₁×t₂×…×t_n, and includes w_n-1.

Now let n approach infinity. Clearly w_n approaches q. It also approaches a set f, which contains all finite products of terms of M, where each finite product multiplies one term from each of the first n rows, for some integer n. Since f contains positive terms, and approaches q, it is absolutely convergent. The sum of finite products, in any order, gives the product of the sums.

Is this theorem ever used? More often than you might think. Here is a beautiful application.