488 Chapter 8 • Infinite Series of Real Numbers
If we collect terms along the northeast-to-southwest diagonals, we have
(a1 + a2 + a3)(b1 + b2 + b3 + b4) =
a 1 b 1 + (a 1 b2 + a2bi) + (a1b3 + a2b2 + a3b1) + (a1b4 + a2b3 + a3b2) + (a2b4 +
a3b3) + a3b4.
Guided by this understanding, we want the series representing the product
o= ak) cz= bk) to represent the sum of all the entries in the infinite matrix:a;i-u·( O,i.-02' q,-ro?,' a,i.-04' a;i-u(·..
,e ,-' ,-' ,-' ,-'
9-20l. <J2V2 a,203 a:;.b4 a2bs · · ·
~ ---- ,-'' ,-''
)J-3'b1 9-st52 q.s-/J3 a3b4 a3bs · · ·
jJ,4'b';' g,<flf2,, a4b3 a4b4 a4bs · · · ·
jvsf5~--asb2 asb3 asb4 asbsAny "product" of I: ak and I: bk must add all the entries in this infinite
matrix. But how can we do that? Having studied conditionally convergent
series, we know that the order of summing can affect the results. The idea
behind the "Cauchy product" series is to add all the entries in this matrix by a
procedure patterned after the one we used in the finite case above. That is, we
add along the northeast-to-southwest diagonals and group these sums together
as terms of a series. More specifically, we make the following definition.00 00
Definition 8.4.l The Cauchy product of I: ak and I: bk is the series
00
I: Ck, where
k=lk=l k=lk
ck= L aibk+l-i = a1bk + a2bk-1 + · · · + ak-1b2 + akb1.
i=l
00 00
For series I: ak and I: bk, the formula for Ck is
k=O k=O
k
ck = L aibk-i = aobk + a1bk-1 + · · · + ak-1b1 + akbo.
i=O
Two important questions are (1) under what conditions are we guaranteed
that the Cauchy product series converges, and (2) when the Cauchy product of
two series converges, does it converge to the product of their sums? Perhaps
surprisingly, absolute convergence plays a key role in answering these questions.
Before proving any theorems, we give an example showing that the Cauchy
product of two convergent series does not necessarily converge.