This preview shows pages 1–2. Sign up to view the full content.
MATH185: Assignment 10
Throughout this homework let
U
be an open subset of
C
:
1. Recall that a sequence of functions
f
n
:
U
→
C
is uniformly Cauchy
on
U
iﬀ
lim
N
→∞
(
sup
{
f
n
(
w
)

f
m
(
w
)

:
n, m
≥
N
and
w
∈
U
}
)
= 0
(a) Show that a sequence of functions
f
n
is Uniformly Cauchy on
U
if and only if there is a function
g
:
U
→
C
such that
f
n
converges
uniformly to
g
on
U
.
(b) Use part (
a
) to show that, if the series
∑
∞
n
=1

f
n
(
z
)

converges
uniformly on
U
then the series
∑
∞
n
=1
f
n
(
z
) converges uniformly
on
U
(Recall that a series converges unformly if it converges to
and moreover the sequence of partial sums converges uniformly
to the limit)
2. The following is a very useful method to show that a series of functions
converges uniformly on
U
:
Suppose that there exists a sequence of positive real numbers
c
n
such
that for
n >>
0 and for all
z
∈
U
we have

f
n
(
z
)
 ≤
c
n
. Show that if
∑
∞
n
=1
c
n
converges then
∑
∞
n
=1
f
n
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '07
 Lim
 Math

Click to edit the document details