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Real Analysis
Notes 1 1 1 1
Therefore, lim f (x) dx = 1 ò f(x) dx = ò f(x) dx = ò lim (f (x))dx . That is, the integral of the limit
n ò
n
n®¥ 2 n®¥
0 0 0 0
is not equal to be limit of the sequence of integrals. In fact, (f ) is not uniformly convergent to f
n
in [0, 1]. This we prove by the contradiction method. If possible, let the sequence be uniformly
1 1
convergent in [0, 1]. Then, for = , there exists a positive integer m such that |f(x) – f(x)| < ,
4 4
for n m and " x [0, l].
nx 1
i.e., 2 < , for n m and V x [0, 1]
e nx 4
1
Choose a positive integer M m such that [0, 1],
M
1
Take n = M and x . We get
M
1 1 e 2
< i.e., M < < 1.
M 4 16
which is a contradiction. Hence (f) is not uniformly convergent in [0, 1].
Now we give the theorems without proof which relate uniform convergence with continuity,
differentiability and integrability of the limit function of a sequence of functions.
Theorem 2: Uniform Convergence and Continuity
If (f ) be a sequence of continuous functions defined on [a, b] and (f ) ® f uniformly on [a, b], then
n n
f is continuous on [a, b].
Theorem 3: Uniform Convergence and Differentiation
Let (f ) be a sequence of functions, each differentiable on [a, b] such that (f(x )) converges for some
n n 0
point x of [a, b]. If (f ) converges uniformly on [a, b] then (f ) converges uniformly on [a, b] to a
0 n n
function f such that
f’(x) = lim f’ (x); x [a, b].
n®¥ n
Theorem 4: Uniform Convergence and Integration
If a sequence (f ) converges uniformly to f on [a, b] and each function f is integrable on [a, b],
n n
then f is integrable on [a, b] and
b b
J f(x) dx = lim J f (x) dx
n
a n®¥ a
14.3 Series of Functions
Just as we have studied series of real numbers, we can study series formed by a sequence of
functions defined on a given set A. The ideas of pointwise convergence and uniform convergence
of sequence of functions can be extended to series of functions.
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