Sequence and Series of Functions
Introduction
Last Updated: January 14, 2025Sequence of real numbers: Consider sequence of real numbers \begin{align} & \{y_{n}\}=\left\{ 1,\frac{1}{2},\frac{1}{3},…,\frac{1}{n},…\right\} \nonumber\\ & \end{align} Then clearly $$\{y_{n}\}\to 0 ~as ~n\to \infty~or~\lim\limits_{n\to \infty}y_{n}=0 $$ And, by the definition of limit of a sequence, for any positive number \(\epsilon~,~\exists \) a positive integer \(k \) such that \begin{align} & |y_{n}-0|
Pointwise Convergence
Last Updated: June 24, 2024Definition: Let \(\bf \{ f_{n}(x)\} \) be a sequence of real velued functions, defined on \(\bf E\subseteq \real \). The sequence \(\bf \{ f_{n}(x)\} \) is said to be \(\fcolorbox{red}{white}{\bf pointwise convergent} \)to a real valued function \(\bf f(x) \), defined on \(\bf E\subseteq \real \), if for any positive number \(\bf \epsilon~,~\exists \)...
