Sequences and Series of Functions
We will cover following topics
Uniform Convergence of Sequence of Functions
A sequence of functions s{fn},n=1,2,3,… uniformly converges to f for a set E of values of x, if, for every ε>0, ∃ an integer N such that |fn(x)−f(x)|<ϵ ∀ n≥N,x∈E.
Uniform Convergence of Series of Functions
A series of function ∑fn(x) converges uniformly on E, if the sequence {Sn} of partial sums defined by Sn(x)=∑nk=1fk(x) converges uniformly on E.
Tests for Uniform Convergence of Series of Functions
Weierstrass M-test
Let (fn) be a sequence of functions defined on D, and suppose that there is a sequence of non-negative numbers (Mn) such that
|fn(x)|≤Mn for all x∈D,n∈NIf the series ∑Mn converges, then both ∑fn and ∑|fn| converge uniformly on D.
Dirichlet Test
Let (un) and (vn) be two sequences of real valued functions on a set D⊆R which satisfy the following conditions:
(i) The sequence (Un) of partial sums of (un) is uniformly bounded, in the sense that there is a constant K such that
|Un(x)|=|∑nk=1uk(x)|≤K for all x∈D,n∈N
(ii) The sequence (vn) is monotonically decreasing on D; that is, vn+1(x) ≤vn(x) for all n∈N and x∈D (iii) vn ù →0 on D.
Then the series ∑unvn is uniformly convergent on D.
Properties of Uniformly Convergent Series of Functions
In all results, we shall suppose that each of the function fn has [a,b] 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 Σfn be a uniformly convergent series of integrable functions in [a,b] and S(x) denotes its sum, then S(x) is integrable in [a,b] and
∫baΣfn(x)dx=∫baS(x)dx=Σ∫bafn(x)dxi.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
ddx(Σfn)=Σf′ni.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
fn(x)=nx1+n2x2,∀x∈R(−∞,∞), n=1,2,3..
[2019, 15M]
2) Let fn(x)=nx(1−x)n, x∈[0,1]. Examine the uniform convergence of {fn(x)} on [0,1].
[2011, 15M]
3) Let fn(x)=xn on −1<x≤1 for n=1,2……. Find the limit function. Is the convergence uniform? Justify your answer.
[2010, 15M]
Uniform Convergence of Series
1) Test the series of functions ∑∞n=1nx1+n2x2 for uniform convergence.
[2015, 15M]
2) Show that the series ∑∞1(−1)n−1n+x2, is uniformly convergent but not absolutely for all real values of x.
[2013, 13M]
3) Show that the series for which the sum of first n terms, fn(x)=nx1+n2x2, 0≤x≤1 cannot be differentiated term-by-term at x=0. What happens at x≠0?
[2011, 15M]
4) Show that if S(x)=∑∞n=11n3+n4x2, then its derivative S′(x)=−2x∑∞n=11n2(1+nx2)2, for all x.
[2011, 20M]
5) Consider the series ∑∞n=0x2(1+x2)n.
Find the values of x for which it is convergent and also the sum function. Is the convergence uniform? Justify your answer.
[2010, 15M]
6) Show that:
limx→1∑∞n=1n2x2n4+x4= ∑∞n=1n2n4+1
Justify all steps of your answer by quoting the theorems you are using.
[2009, 15M]
7) Discuss the convergence of the series x2+1.32.4x2+1.3.52.4.6x3+….., x>0.
[2008, 12M]
8) Test uniform convergence of the series ∑∞n=1sinnxnp, where p>0.
[2002, 20M]