Math 409-502
Harold P. Boas
boas@tamu.edu
Results of the third examination
Good job!
Here are the scores:
95 94
85 85
94
80
92
79
92 91
78 78
90
77
90 89
76 75
88
66
88
34
Reminder: the comprehensive final examination will be held in this room on Tuesday, December 14, from 8:00–10:00 AM.
Math 409-502
December 3, 2004 — slide #2
Things can go wrong in the limit
Continuity. On the(
interval [0, 1], let f n (x) = x1/n .
1, if x 6= 0,
Then lim f n (x) =
n→∞
0, if x = 0,
so the limit of these continuous functions is discontinuous.
³
´0
Derivatives. Let gn (x) = n1 x n . Then lim gn (x) = 0 when 0 ≤ x ≤ 1, but lim gn0 (x) =
n→∞
n→∞
(
0, if x 6= 1,
lim x n−1 =
n→∞
1, if x = 1.
Here (derivative of the limit) 6= (limit of the derivatives).
(
1
2n+1 , if 2n+1
≤ x ≤ 21n ,
Integrals. Let h n (x) =
0,
otherwise.
R1
R1
Then 0 lim hn (x) dx = 0, but lim 0 hn (x) dx = 1.
n→∞
n→∞
Here (integral of the limit) 6= (limit of the integrals).
Math 409-502
December 3, 2004 — slide #3
Uniform convergence
A sequence of functions { f n }∞
n=1 converges uniformly on an interval to a function f if for every e > 0 there is an N such that | f n (x) − f (x)| < e for all x whenever n > N.
“Uniform” means that N can be chosen to be independent of x.
Example. Let f n (x) = sin(x + n1 ). Then the sequence of functions f n (x) converges uniformly to
the function f (x) = sin(x) on the unbounded interval (−∞, ∞).
For suppose e > 0 is given. By the mean-value theorem, sin(x + n1 ) − sin(x) = n1 cos(c) for
some c depending on x, so | f n (x) − f (x)| ≤ n1 . Therefore | f n (x) − f (x)| < e when n > 1/e. So
we can take N = 1/e, which is independent of x.
Math 409-502
December 3, 2004 — slide #4
Theorems about uniform convergence
1. If a sequence of continuous functions f n converges uniformly on an interval to a function f ,
then the limit function f is continuous.
2. If a sequence of differentiable functions f n converges (uniformly) on an interval to a function f , and if the sequence of derivatives f n0 converges uniformly to a function g, then the limit
function f is differentiable, and f 0 = g.
3. If a sequence of integrable functions f n converges uniformly on an interval [a, b] to a funcRb
Rb
tion f , then f is integrable and a f (x) dx = lim a f n (x) dx.
n→∞
(This is a stronger theorem than the one in the book.)
In all three cases, the word “sequence” can be replaced by the word “series” (because convergence of a series means convergence of the sequence of partial sums).
Math 409-502
December 3, 2004 — slide #5
Homework
Read sections 22.1–22.5, pages 305–318.
Math 409-502
December 3, 2004 — slide #6