Uniform Convergence of Sequence of Functions

A sequence of functions $s {f_{n}}, n = 1, 2, 3, \dots$ uniformly converges to $f$ for a set $E$ of values of $x$ , if, for every $ε > 0$ , $\exists$ an integer $N$ such that $| f_{n} (x) - f (x) | < ϵ$ $\forall$ $n \geq N, x \in E$ .

Uniform Convergence of Series of Functions

A series of function $\sum f_{n} (x)$ converges uniformly on $E$ , if the sequence ${S_{n}}$ 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 $(f_{n})$ be a sequence of functions defined on $D,$ and suppose that there is a sequence of non-negative numbers $(M_{n})$ such that

| f_{n} (x) | \leq M_{n} for all x \in D, n \in N

If the series $\sum M_{n}$ converges, then both $\sum f_{n}$ and $\sum | f_{n} |$ converge uniformly on $D$ .

Dirichlet Test

Let $(u_{n})$ and $(v_{n})$ be two sequences of real valued functions on a set $D \subseteq R$ which satisfy the following conditions: (i) The sequence $(U_{n})$ of partial sums of $(u_{n})$ is uniformly bounded, in the sense that there is a constant $K$ such that $| U_{n} (x) | = | \sum_{k = 1}^{n} u_{k (x)} | \leq K for all x \in D, n \in N$
(ii) The sequence $(v_{n})$ is monotonically decreasing on $D;$ that is, $v_{n + 1} (x)$ $\leq v_{n} (x)$ for all $n \in N$ and $x \in D$ (iii) $v_{n} \overset{ù}{\to} 0$ on $D$ .
Then the series $\sum u_{n} v_{n}$ is uniformly convergent on $D$ .

Abel Test

Suppose the sequences $(u_{n})$ and $(v_{n})$ of real valued functions defined on $D \subseteq R$ satisfy the following conditions:
(i) $\sum u_{n}$ converges uniformly on $D$ .
(ii) The sequence $(v_{n})$ 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 $Σ f_{n}$ 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} Σ f_{n} (x) d x = \int_{a}^{b} S (x) d x = Σ \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} (Σ f_{n}) = Σ f_{n}^{'}

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

$f_{n} (x) = \frac{n x}{1 + n^{2} x^{2}}, \forall x \in R (- \infty, \infty)$ , $n = 1, 2, 3..$

[2019, 15M]

2) Let $f_{n} (x) = n x (1 - x)^{n}$ , $x \in [0, 1]$ . Examine the uniform convergence of ${f_{n} (x)}$ on $[0, 1]$ .

[2011, 15M]

3) Let $f_{n} (x) = x^{n}$ on $- 1 < x \leq 1$ for $n = 1, 2 \dots \dots$ . Find the limit function. Is the convergence uniform? Justify your answer.

[2010, 15M]

Uniform Convergence of Series

1) Test the series of functions $\sum_{n = 1}^{\infty} \frac{n x}{1 + n^{2} x^{2}}$ for uniform convergence.

[2015, 15M]

2) Show that the series $\sum_{1}^{\infty} \frac{(- 1)^{n - 1}}{n + x^{2}}$ , 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, $f_{n} (x) = \frac{n x}{1 + n^{2} x^{2}}$ , $0 \leq x \leq 1$ cannot be differentiated term-by-term at $x = 0$ . What happens at $x \neq 0$ ?

[2011, 15M]

4) Show that if $S (x) = \sum_{n = 1}^{\infty} \frac{1}{n^{3} + n^{4} x^{2}}$ , then its derivative $S^{'} (x) = - 2 x \sum_{n = 1}^{\infty} \frac{1}{n^{2} {(1 + n x^{2})}^{2}}$ , for all $x$ .

[2011, 20M]

5) Consider the series $\sum_{n = 0}^{\infty} \frac{x^{2}}{{(1 + x^{2})}^{n}}$ .

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:

$lim_{x \to 1} \sum_{n = 1}^{\infty} \frac{n^{2} x^{2}}{n^{4} + x^{4}}$ = $\sum_{n = 1}^{\infty} \frac{n^{2}}{n^{4} + 1}$

Justify all steps of your answer by quoting the theorems you are using.

[2009, 15M]

7) Discuss the convergence of the series $\frac{x}{2} + \frac{1.3}{2.4} x^{2} + \frac{1.3 .5}{2.4 .6} x^{3} + \dots . .$ , $x > 0$ .

[2008, 12M]

8) Test uniform convergence of the series $\sum_{n = 1}^{\infty} \frac{\sin n x}{n^{p}}$ , where $p > 0$ .

[2002, 20M]

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