Summary: 1. Power series.
Definition 1.1. Suppose c is a sequence in C. (c will be a coefficient sequence.)
We set
M(c, r) = sup{|cn|rn
: n N} whenever 0 r <
and we let
R(c) = sup{r [0, ) : M(c, r) < }.
Note that M(c, r) and R(c) could equal . We call R(c) the radius of conver-
gence of c for reasons which will shortly become apparent.
We repeatedly use the following estimate:

n=N
|cn||z - z0|n
M(c, r)
|z - z0|
r
N r
r - |z - z0|
whenever |z - z0| < r < R(c) and N N.
(1)

Source: Allard, William K. - Department of Mathematics, Duke University

Collections: Mathematics