1550251515-Classical_Complex_Analysis__Gonzalez_

206 Chapter4

term anzn is a continuous function everywhere (in particular, at zo), the

continuity of f(z) at z 0 is a consequence of Theorem 4.18. Because Zo was

an arbitrary point of lzl < R, it follows that f is a continuous function

in that region.

The continuity property of a power series may be expressed as follows:

00 00

z~zo~ lim """"' anzn = L anz~, lzol < R

n=O n=O

Note In Section 8.1 we prove that the function f(z) defined by the power

series is not only continuous but analytic in lz I < R.

Theorem 4.20 Suppose that an -=/= 0 for all n. If

lim I an I-R

n ..... oo an+l

( 4.11-5)

exists (finite or infinite), then R is the radius of convergence of the series

Eanzn.

Proof It suffices to apply the second form of the ratio test [Section 4.9,

part (cl)] to the series E anzn and make use of ( 4.11-5). We find that

if R = oo

if 0 < R < oo

if R = O,z-=/= 0

so the series E anzn converges absolutely for every z E C if R = oo, for

lzl < R if 0 < R < oo, and diverges for every z, except z = 0, if R = 0.

Corollary 4.6 If limn_. 00 lan+i/anl exists, then

limsup ~ = lim \ an+i \ (4.11-6)
n~oo n~oo an
This follows at once from (4.11-4) and (4.11-5). Of course, (4.11-6) can
be applied to any complex sequence {an} provided that the limit on the
right exists.
Remark Rule ( 4.11-5) to fine the radius of convergence is usually easier
to apply than ( 4.11-4), particularly in t,he case of a power series with co-
efficients an containing factorials or polynomials in n. However, ( 4.11-4)
is a more general rule since ( 4.11-5) is applicable only when the ordinary
limit of the ratio lanl/lan+1 I exists.
In case that limit fails to exist, all that can be asserted is that the radius
of convergence R lies in the interval [1/ Li, 1/ L2], where Li is the limit