Page 197 - DMTH401_REAL ANALYSIS

Unit 14: Sequences and Series of Functions

4. Let (f ) be a sequence of functions, each differentiable on [a, b] such that (f (x )) converges for Notes
n n 0
some point x of [a, b]. If (f ) converges uniformly on [a, b] then (f ) converges uniformly on
0 n n
[a, b] to a function f such that .............................
5. If a sequence (f ) converges uniformly to f on [a, b] and each function f is integrable on
n n
[a, b], then f is integrable on [a, b] and .............................
14.4 Summary

 In this unit you have learnt how to discuss the pointwise and uniform convergence of
sequences and series of functions. Sequence of functions is defined and pointwise
convergence of the sequence of functions has been discussed. We say that a sequence of
functions (f ) is pointwise convergent to f on a set A if given a number e > 0, there is a
n
positive integer m such that
|f (x) – f(x)| <  for n  m, x  A.
n
m in general depends on  and the point x under consideration. If it is possible to find m
which depends only on s and not the point x under consideration, then (f ) is said to be
n
uniformly convergent are f on A. Cauchy’s criteria for uniform convergence are discussed.
Also in this section you have seen that if the sequence of functions (f ) is uniformly
n
convergent to a function f on [a, b] and each f ’ is continuous or integrable, then f is also
n
continuous or integrable on [a,b]. Further it has been discussed that if (f ) is a sequence of
n
functions, differentiable on [a,b] such that (f (x,)) converges for some point x of [a, b] and
n 0
if (f ) converges uniformly on [a, b], then (f ) converges uniformly to a differentiable
n n
function f such that f’(x) = lim f’ (x); x [a, b].
n®¥ n
 Finally pointwise and uniform convergence of series of functions is given. The series of
functions is said to be pointwise or uniformly convergent on a set A according as the
sequence of partial sums (s ) of the series is pointwise or uniformly convergent on A.
n
14.5 Keywords

Uniform Convergence and Continuity: If (f ) be a sequence of continuous functions defined on
n
[a, b] and (f ) ® f uniformly on [a, b], then f is continuous on [a, b].
n
Uniform Convergence and Differentiation: Let (f ) be a sequence of functions, each differentiable
n
on [a, b] such that (f (x )) converges for some point x of [a, b]. If (f ) converges uniformly on [a, b]
n 0 0 n
then (f ) converges uniformly on [a, b] to a function f such that
n
f’(x) = lim f’ (x); x [a, b].
n®¥ n
Uniform Convergence and Integration: If a sequence (f ) converges uniformly to f on [a, b] and
n
each function f is integrable on [a, b], then f is integrable on [a, b] and
n
b b
J f(x) dx = lim J f (x) dx
a n®¥ a n
14.6 Review Questions

1. Examine which of the following sequences of functions converge pointwise
(i) f (x) = sin nx, –¥ < x < + ¥
n
mx
(ii) f (x) = 2 , – ¥< x < + ¥
2
n 1 n x
+
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