MAT 401 HOMEWORK 2 DUE FRI 2/27.
Recall the following definition from class.
Definition 0.1. Let A ⊂ C. Given a sequence of functions fn : A −→ C
and another function f on A, we say that fn converges to f uniformly if,
for all > 0, there exists N > 0 such that
|f (x) − fn (x)| < for all n ≥ N and all x ∈ A.
We have proven that if fn → f uniformly on A, and if each fn is continuous, then f is continuous on A.
P
We say that a series of functions
fn is converges uniformly to f if
the sequence sn of partial sums of the series converges uniformly to f . We
proved the following.
Theorem 0.2 (Weierstrass M -test). Given functions fn : A −→ C, suppose
there exists a sequence
M1 , M2 , . . .
of nonnegative real constants satisfying:
∞
X
•
Mn < ∞
n=1
• |fn (x)| ≤ Mn for all x ∈ A.
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Then the series f (x) = ∞
n=1 fn (x) converges absolutely and uniformly on
A. In particular, if each fn is continuous, then f is continuous function.
Exercises:
(1) Suppose fn : [a, b] −→ R is a sequence of continuous functions converging uniformly to a function f . Prove that
Z b
Z b
lim
fn (x)dx =
f (x)dx.
n→∞ a
x2
a
x3
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(2) Let f (x) = 1 + x + 2! + 3! + · · · = ∞
n=0
M -test to prove that f is continuous on R.
xn
n! .
Use the Weierstrass