Infinite Products, The Product of Two Infinite Series

The Product of Two Infinite Series

As you recall, each series implies a sequence of partial sums.  Let u hold the partial sums of s and let v hold the partial sums of t.  Thus u_n is the sum of the first n terms of s, and v_n is the sum of the first n terms of t.

Let w be the term by term product of u and v.  Thus w_n is the first n terms of s times the first n terms of t.  The limit of w is the limit of u times the limit of v, which is the sum of s times the sum of t.  Let's evaluate the limit of w.

The n^th term of w is the sum of s_it_j as i and j run from 1 to n.  Let M be a 2 dimensional matrix with M_i,j = s_it_j.  Now w_n is the sum of the upper left entries of M, n rows and n columns.  As n increases we cover the entire array.  The limit of w is the limit of M, as a 2 dimensional series.  As long as M converges, the limit of w is the limit of s times the limit of t, which is what we want to prove.

The terms of M are nonnegative, so we can add across rows, and then add the rows together.  If this nested sum converges then M converges.

Let q be the sum over t.  Thus the k^th row of M sums to s_k×q.  Add up all the rows, and we are adding the series s, scaled by q.  This is convergent, and its limit is the sum of s multiplied by q, or the sum of s times the sum of t.

either s or t could be a nonpositive series, instead of a nonnegative series; the theorem still holds.  As long as the entries of M have the same sign, positive or negative, we have absolute convergence, and the sum of M is the limit of w, is the sum of s*t.

Next let s and t be absolutely convergent series.  Each series could contain both positive and negative terms.  Our proof runs into a snag when we evaluate the sum of M.  If has positive and negative terms; does it converge?  Write M as the sum of two matrices, one with the positive terms and one with the negative terms.  Each is dominated by a matrix where all the terms of s and t are made positive, and that matrix converges.  Thus the positive and negative "halves" of M are absolutely convergent, and M is absolutely convergent.  Once again the product s*t converges to the sum of s times the sum of t.

Use induction to generalize this to a finite product of absolutely convergent series.  For instance, the product of the three series s t and u is the sum of s_it_ju_k for all i j and k > 0, and the limit of this 3 dimensional series is equal to the sum of s times the sum of t times the sum of u.