Sequences and Series of Functions
We will cover following topics
Uniform Convergence of Sequence of Functions
A sequence of functions uniformly converges to for a set of values of , if, for every , an integer such that .
Uniform Convergence of Series of Functions
A series of function converges uniformly on , if the sequence of partial sums defined by converges uniformly on .
Tests for Uniform Convergence of Series of Functions
Weierstrass M-test
Let be a sequence of functions defined on and suppose that there is a sequence of non-negative numbers such that
If the series converges, then both and converge uniformly on .
Dirichlet Test
Let and be two sequences of real valued functions on a set which satisfy the following conditions:
(i) The sequence of partial sums of is uniformly bounded, in the sense that there is a constant such that
(ii) The sequence is monotonically decreasing on that is, for all and (iii) on .
Then the series is uniformly convergent on .
Properties of Uniformly Convergent Series of Functions
In all results, we shall suppose that each of the function has as its domain.
i) The sum of a uniformly convergent series of continuous functions is continuous.
ii) The sum of a uniformly convergent series of integrable functions is integrable and the integral of the sum is equal to the sum of the series of integrals of the functions.
Thus if be a uniformly convergent series of integrable functions in and denotes its sum, then is integrable in and
i.e., the series is term by term integrable.
iii) If $\Sigma f_{n}$ is a point-wise convergent series of derivable functions with continuous derivatives and the series $\Sigma f_{n}^{\prime}$ of derivatives is uniformly convergent, then
i.e., the derivative of the sum is equal to the sum of the derivatives. Thus, the term-by-term differentiation of the series is valid.
PYQs
Uniform Convergence of Sequence
1) Discuss the uniform convergence of
,
[2019, 15M]
2) Let , . Examine the uniform convergence of on .
[2011, 15M]
3) Let on for . Find the limit function. Is the convergence uniform? Justify your answer.
[2010, 15M]
Uniform Convergence of Series
1) Test the series of functions for uniform convergence.
[2015, 15M]
2) Show that the series , is uniformly convergent but not absolutely for all real values of .
[2013, 13M]
3) Show that the series for which the sum of first terms, , cannot be differentiated term-by-term at . What happens at ?
[2011, 15M]
4) Show that if , then its derivative , for all .
[2011, 20M]
5) Consider the series .
Find the values of for which it is convergent and also the sum function. Is the convergence uniform? Justify your answer.
[2010, 15M]
6) Show that:
=
Justify all steps of your answer by quoting the theorems you are using.
[2009, 15M]
7) Discuss the convergence of the series , .
[2008, 12M]
8) Test uniform convergence of the series , where .
[2002, 20M]