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
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 Fall '07
 Lim
 Math, Calculus, converges, uniformly Cauchy

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