Sequences and Series of Functions
We will cover following topics
Uniform Convergence of Sequence of Functions
A sequence of functions \(s\left\{f_{n}\right\}, n=1,2,3, \dots\) uniformly converges to \(f\) for a set \(E\) of values of \(x\), if, for every \(\varepsilon > 0\), \(\exists\) an integer \(N\) such that \(\vert f_{n}(x)-f(x) \vert <\epsilon\) \(\forall\) \(n \geq N, x \in E\).
Uniform Convergence of Series of Functions
A series of function \(\sum f_{n}(x)\) converges uniformly on \(E\), if the sequence \(\left\{S_{n}\right\}\) of partial sums defined by \(S_{n}(x) = \sum_{k=1}^{n} f_{k}(x)\) converges uniformly on \(E\).
Tests for Uniform Convergence of Series of Functions
Weierstrass M-test
Let \(\left(f_{n}\right)\) be a sequence of functions defined on \(D,\) and suppose that there is a sequence of non-negative numbers \(\left(M_{n}\right)\) such that
\[\vert f_{n}(x)\vert \leq M_{n} \quad \text { for all } x \in D, n \in \mathbb{N}\]If the series \(\sum M_{n}\) converges, then both \(\sum f_{n}\) and \(\sum\vert f_{n} \vert\) converge uniformly on \(D\).
Dirichlet Test
Let \(\left(u_{n}\right)\) and \(\left(v_{n}\right)\) be two sequences of real valued functions on a set \(D \subseteq \mathrm{R}\) which satisfy the following conditions:
(i) The sequence \(\left(U_{n}\right)\) of partial sums of \(\left(u_{n}\right)\) is uniformly bounded, in the sense that there is a constant \(K\) such that
\(\vert U_{n}(x)\vert =\vert \sum_{k=1}^{n} u_{k(x)}\vert \leq K \text { for all } x \in D, n \in \mathbb{N}\)
(ii) The sequence \(\left(v_{n}\right)\) is monotonically decreasing on \(D ;\) that is, \(v_{n+1}(x)\) \(\leq v_{n}(x)\) for all \(n \in \mathbb{N}\) and \(x \in D\) (iii) \(v_{n} \stackrel{\text { รน }}{\rightarrow} 0\) on \(D\).
Then the series \(\sum u_{n} v_{n}\) is uniformly convergent on \(D\).
Abel Test
Suppose the sequences \(\left(u_{n}\right)\) and \(\left(v_{n}\right)\) of real valued functions defined on \(D \subseteq \mathrm{R}\) satisfy the following conditions:
(i) \(\sum u_{n}\) converges uniformly on \(D\).
(ii) The sequence \(\left(v_{n}\right)\) is uniformly bounded and monotonically decreasing on \(D\). Then the series \(\sum u_{n} v_{n}\) is uniformly convergent on \(D\).
Properties of Uniformly Convergent Series of Functions
In all results, we shall suppose that each of the function \(f_{n}\) 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 \(\Sigma f_{n}\) 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
\[\int_{a}^{b} \Sigma f_{n}(x) d x=\int_{a}^{b} S(x) d x=\Sigma \int_{a}^{b} f_{n}(x) d x\]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
\[\frac{d}{d x}\left(\Sigma f_{n}\right)=\Sigma f_{n}^{\prime}\]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
\(f_n(x)=\dfrac{nx}{1+n^2x^2},\forall x\in R(-\infty,\infty)\), \(n=1,2,3..\)
[2019, 15M]
2) Let \(f_{n}(x)=n x(1-x)^{n}\), \(x \in[0, 1]\). Examine the uniform convergence of \(\left\{f_{n}(x)\right\}\) on \([0, 1]\).
[2011, 15M]
3) Let \(f_{n}(x)=x^{n}\) on \(-1 < x \leq 1\) for \(n=1,2 \ldots \ldots\). Find the limit function. Is the convergence uniform? Justify your answer.
[2010, 15M]
Uniform Convergence of Series
1) Test the series of functions \(\sum_{n=1}^{\infty} \dfrac{n x}{1+n^{2} x^{2}}\) for uniform convergence.
[2015, 15M]
2) Show that the series \(\sum_{1}^{\infty} \dfrac{(-1)^{n-1}}{n+x^{2}}\), 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, \(f_{n}(x)=\dfrac{n x}{1+n^{2} x^{2}}\), \(0 \leq x \leq 1\) cannot be differentiated term-by-term at \(x=0\). What happens at \(x \neq 0\)?
[2011, 15M]
4) Show that if \(S(x)=\sum_{n=1}^{\infty} \dfrac{1}{n^{3}+n^{4} x^{2}}\), then its derivative \(S^{\prime}(x)=-2 x \sum_{n=1}^{\infty} \dfrac{1}{n^{2}\left(1+n x^{2}\right)^{2}}\), for all \(x\).
[2011, 20M]
5) Consider the series \(\sum_{n=0}^{\infty} \dfrac{x^{2}}{\left(1+x^{2}\right)^{n}}\).
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:
\(\lim_{x \to 1} \sum_{n=1}^{\infty} \dfrac{n^2x^2}{n^4+x^4}\)= \(\sum_{n=1}^{\infty} \dfrac{n^2}{n^4+1}\)
Justify all steps of your answer by quoting the theorems you are using.
[2009, 15M]
7) Discuss the convergence of the series \(\dfrac{x}{2}+\dfrac{1.3}{2.4} x^{2}+\dfrac{1.3 .5}{2.4 .6} x^{3}+\ldots . .\), \(x>0\).
[2008, 12M]
8) Test uniform convergence of the series \(\sum_{n=1}^{\infty} \dfrac{\sin n x}{n^{p}}\), where \(p>0\).
[2002, 20M]