1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series 191

By assumption, the series

00 00

and

are convergent. The series of the products of their terms as in ( 4.8-2) is

( 4.8-3)

Let O"n be the sum of the first n terms of ( 4.8-3), and let a be the greatest

index of the factors jajj, (3 the greatest index of the factors lbkl contained

in O"n. Then the product

contains all the terms in O"n (and possibly others). Hence we have
O"n ::::; Co:Df3 ::::; A' B'
Since the sequence { O" n} of positive terms is increasing and bounded above,
it has a finite limit (Exercises 4.1, problem 17). Thus ( 4.8-3) converges,
i.e., ( 4.8-2) converges absolutely, and by Theorems 4.8 and 4.9 we can
rearrange and group its terms as follows without altering its sum:

ai b1 + ( ai b2 + a2b1 + a2b2) + ( aib3 + a2b3 + a3b1 + a3b2 + a3b3) + · · · ( 4.8-4)

In the first term the subindices are 1, the second term contains those re-
maining with subindices ~ 2, the third those remaining with subindices ::::;

3, and so on. If Sn denotes the partial sum of the series ( 4.8-4), and An, Bn

the partial sums of the series :Z:::~ an, :Z:::~ bn, respectively, we have

... '

Hence S = limn-->oo Sn = AB.

Corollary 4.3 If :Z:::~ an and :Z:::~ bn are both absolutely convergent, then

where Cn = aibn + a2bn-l + · · · + anb1.

Proof This amounts to the following rearrangement of the terms of ( 4.8-2),

called Cauchy's rule:

( 4.8-5)

in which the terms with a constant sum of indices are put together. The
arrangement ( 4.8-5) proves to be the most convenient way of expressing
the product of the two series, especially in the theory of power series.