Pointwise Convergent and Uniformly Convergent Series of Functions

# Pointwise Convergent and Uniformly Convergent Series of Functions

Recall from the Pointwise Convergence of Sequences of Functions page 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 $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then:

(1)
\begin{align} \quad \mid f_n(x) - f(x) \mid < \epsilon \end{align}

On the Uniform Convergence of Sequences of Functions page we said that a sequence of functions $(f_n(x))_{n=1}^{\infty}$ with common domain $X$ is said to be uniformly convergent if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ then for all $x \in X$ we have that:

(2)
\begin{align} \quad \mid f_n(x) - f(x) \mid < \epsilon \end{align}

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 $(-1, 1)$:

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

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.