Uniform Convergence of Series of Functions

Pointwise Convergent and Uniformly Convergent Series of Functions

Recall that a sequence of functions $(f_{n}(x))_{n=1}^{\infty }$ with common domain $X$ is said to be pointwise convergent if for all $x\in X$ and for all $\varepsilon >0$ there exists an $N\in \mathbb {N}$ such that if $n\geq N$ then:

${\begin{aligned}\quad \left|f_{n}(x)-f(x)\right|<\varepsilon \end{aligned}}$

Also recall that a sequence of functions $(f_{n}(x))_{n=1}^{\infty }$ with common domain $X$ is said to be uniformly convergent if for all $\varepsilon >0$ there exists an $N\in \mathbb {N}$ such that if $n\geq N$ then for all $x\in X$ we have that:

${\begin{aligned}\quad \left|f_{n}(x)-f(x)\right|<\varepsilon \end{aligned}}$

We will now extend the concept of pointwise convergence and uniform convergence to series of functions.

Definition: Let $(f_{n}(x))_{n=1}^{\infty }$ be a sequence of functions with common domain $X$. The corresponding series $\displaystyle {\sum _{n=1}^{\infty }f_{n}(x)}$ is said to be Pointwise Convergent to the sum function $f(x)$ if the corresponding sequence of partial sums $(s_{n}(x))_{n=1}^{\infty }$ (where $\displaystyle {s_{n}(x)=\sum _{k=1}^{n}f_{n}(x)=f_{1}(x)+f_{2}(x)+...+f_{n}(x)}$) is pointwise convergent to $f(x)$.

For example, consider the following sequence of functions defined on the interval $(-1,1)$:

${\begin{aligned}\quad (f_{n}(x))_{n=1}^{\infty }=(x^{n-1})_{n=1}^{\infty }=(1,x,x^{2},x^{3},...)\end{aligned}}$

We now that this series converges pointwise for all $x\in (0,1)$ since the result series $\sum _{n=1}^{\infty }x^{n-1}$ is simply a geometric series to the sum function $\displaystyle {f(x)={\frac {1}{1-x}}}$.

Definition: Let $(f_{n}(x))_{n=1}^{\infty }$ be a sequence of functions with common domain $X$. The corresponding series $\displaystyle {\sum _{n=1}^{\infty }f_{n}(x)}$ is said to be Uniformly Convergent to the sum function $f(x)$ if the corresponding sequence of partial sums is uniformly convergent to $f(x)$.

The geometric series given above actually converges uniformly on $(-1,1)$, though, showing this with the current definition of uniform convergence of series of functions is laborious. We will soon develop methods to determine whether a series of functions converges uniformly or not without having to brute-force apply the definition for uniform convergence for the sequence of partial sums.

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• Pointwise Convergent and Uniformly Convergent Series of Functions, mathonline.wikidot.com under a CC BY-SA license