1550251515-Classical_Complex_Analysis__Gonzalez_

562 Chapter^8

and write

00

Note Any theorem which states that if a sequence converges to a value,
then some average of the sequence converges to the same value, is called
an Abelian theorem. The name comes from Abel's limit theorem, which is
a theorem of this sort since

for lzl < 1, so that the series E:=o anzn may be thought as a weighted

average of the sequence Sn = a 0 + a 1 + · · ·+an of the partial sums of 2: an,
the weights being 1, z, z^2 , •••• Similarly, property 15 of Theorem 4.1 is
an Abelian theorem.
The converse of an Abelian theorem is false in general. However, it may
become true by adding some suitable condition. The resulting theorem is
called 1'auberian, honoring A. Tauber, who first proved a simple correct
converse of Abel's limit theorem [36]. Such a supplementary condition is
called a Tauberian condition.

Theorem 8.16 (Tauber's 'Theorem). Let f(z) = E:=o anzn for lzJ < 1,
and suppose that

z-+l lim f(z) = S

zEu

rf denoting the same sector as in Theorem 8.15. If an = o(l/n) (i.e., if

limn-+oo nan = 0), then the series 2: an converges to S.

Proof H will do to show that

oo N

Lanzn-Lan~ 0 (8.7-7)

n=O n=O

as z ~ 1 (z E rf), where N = [1/(1-lzl)], the bracket denoting the largest
integer not exceeding 1/(1 - Jzl).
Alternatively, we may write (8.7-7) as follows:

oo N

L anzn - Lan(l-zn) ~ 0 (8.7-8)

n=N+l n=O