8.5 Series of Products 501

SQUARE SUMMABLE SEQUENCES

We are seeking conditions on the series I.: ak and I.: bk, weaker than abso-

=

lute convergence, that will guarantee that I.: akbk converges. That is, we seek

k=l

conditions that guarantee the existence of the "dot product" of the sequences

{ak} and {bk}, as defined in Equation (19) above. With the help of the Cauchy-

Schwarz inequality we shall show that "square summability,'' as defined below,

is sufficient.

Definition 8.5.15 We shall call a sequence { xk} of real numbers summable^7
= =
if I.: Xk converges, absolutely summable if I.: lxkl converges, and square
k=l k=l

summable if I.: x~ converges.
k=l

Absolute summability is stronger than both summability and square summa-
bility (see Exercise 13). However, summability and square summability are not
comparable; that is , neither is stronger than the other (see Exercise 14).

The following theorem shows that square summability of both { ak} and
{bk} is enough to guarantee the convergence of I.: akbk· It is the condition we
have been seeking.

Theorem 8.5.16 If {ak} and {bk} are square summable sequences, then

I.: = akbk converges absolutely.
k=l

Proof. Suppose {ak} and {bk} are square summable and denote the partial
n n
sums of their associated series by An = I.: ak and Bn = I.: bk. Consider a fixed
k=l k=l

7. These are definitions for temporary convenience only, since the term "summable" has a
   more refined definition in higher analysis.