Assignment # 1 MTH 631 Spring 2022 Maximum Marks: 20 Due Date: 13 June, 2022

Question no. 1

SOLUTION:-

Point wise convergence:

Suppose that {F_n} is a sequence of function on D and the sequence of values { F_n (x)} converges for each x in some subset S of D. Then we say that {F_n} converges point wise on S to the limit function F, defined by

$F\left( x \right) = {\lim }\limits_{x \to \infty } {F_n}(x),\,\,\,\,\,\,x \in S.$

Uniform convergence:

A sequence {F_n} of functions defined on a set S converges uniformly to the limit function F on S if

${\lim }\limits_{x \to \infty } \left\| {{F_n} - F} \right\|s = 0$

Thus, {F_n} converges uniformly to F on S if for each $\in > 0$ thereis an integer N such that

$\left\| {{F_n} - F} \right\|s < \in \,\,\,\,\,if\,\,\,n \ge N.$

Point wise convergence:

Given that:-

Function:

${f_n}\left( x \right) = \frac{1}{{nx}} - \frac{{nx}}{{nx + 1}}$

So,

Applying both side limit.

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = {\lim }\limits_{n \to \infty } \left[ {\frac{1}{{nx}} - \frac{{nx}}{{nx + 1}}} \right]$

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = {\lim }\limits_{n \to \infty } \frac{1}{{nx}} - {\lim }\limits_{n \to \infty } \frac{{nx}}{{nx + 1}}$

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = \frac{1}{\infty } - {\lim }\limits_{n \to \infty } \frac{{nx}}{{n\left[ {x + \frac{1}{n}} \right]}}$

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = 0 - {\lim }\limits_{n \to \infty } \frac{x}{{\left[ {x + \frac{1}{n}} \right]}}$

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = \frac{x}{{\left[ {x + \frac{1}{\infty }} \right]}}$

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = 0 - \frac{x}{{\left[ {x + 0} \right]}}$

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = 0 - \frac{x}{x}$

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = 0 - 1$

${\lim }\limits_{n \to \infty } {f_n}\left( x \right) = - 1$

Hence,

Sequence {F_n} converges point wise to for (0,1) $\in R$

A sequence {F_n} of a function define on a set S converges uniformly to the limit function F on S if.

${\lim }\limits_{n \to \infty } \left\| {{F_n} - F} \right\|s = 0$

So,

$= {\lim }\limits_{n \to \infty } \left\| {\frac{1}{{nx}} - \frac{{nx}}{{nx + 1}} + 1} \right\|$

$\begin{array}{l} = - 1 + 1\\ = 0 \end{array}$

Hence,

So, {F_n} is uniformly converges.

SOLUTION:-

Cauchy’s criterion for uniform convergence for series

A series of function $\sum {f_n}$ defined on [a, b] converges uniformly on [a, b] if and only if for every $\in > 0$ and for all $x \in \left[ {a,b} \right]$ , there exist an integer N such that

${f_{n + 1}}(x) + {f_{n + p}}(x)| < \in ,\,\,\,n \ge N,p \ge 1.$

Weierstrass’s test:

A series of functions $\sum {f_n}$ will converge uniformly and absolutely on [a, b] if there exists a convergent series $\sum {M_n}$ of positive numbers such that for all $x \in \left[ {a,b} \right]$ $\left| {{f_n}(x)} \right| \le {M_n}\,\,for\,\,all\,\,n$