1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 561

All three series converge for z = 1, so they converge absolutely for

JzJ < 1. Since

h(z) = f(z)g(z)

for JzJ < 1, we have

lim h(z) = lim f(z) · lim g(z)

z->l z->l z->1

zEu zEu zEu

or C = AB by Theorem 8.15.

As we have seen, Theorem 8.15 (sometimes called Abel's limit theorem)

means that.

(8.7-6)

provided that the series I.:::"=o an converges. However, it may happen that
the limit on the left-hand side of (8. 7-6) exists for certain divergent series
I.:::"=o an. This leads to the following definition.

Definition 8.1 If lim z->1 I.:::o anzn exists (finite), it is said that the
zEu
series I.:::"=o an is Abel summable (briefly, A-summable), and that it has
Abel-sum S.
Abel's limit theorem shows that a convergent series with sum S (in the
ordinary sense) has Abel-sum S. In other words, summability in the Abel
sense is consistent with the ordinary definition of sum of a convergent series.
But there are some nonconvergent series that are A-summable.

Example We have

• 1
  
  
  • = ~(-ltzn
    l+z Lt n=O
  
  
  
  

for JzJ < 1

The series I.:::"= 0 (-l)n = 1 - 1+1 - 1 + · · · is divergent, but

1 1.

lim --·-= -

Z->1 zEu 1 + Z 2

Hence I.:::"= 0 (-lt is Abel summable and its Abel-sum is^1 / 2 • We write

00

L(-lt =^1 / 2 (A)

n=O

In general, if the sum of the series I.:::"=o an in some new sense (say, in the

B sense) is S, we say that the series is B-summable, call S the B-sum,