Uniform Convergence of 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.

Cauchy's Uniform Convergence Criterion for Series of Functions

If we have a sequence of functions $(f_{n}(x))_{n=1}^{\infty }$ with common domain $X$ then the corresponding series of functions $\displaystyle {\sum _{n=1}^{\infty }f_{n}(x)}$ is said to be uniformly convergent if the corresponding sequence of partial sums $(s_{n}(x))_{n=1}^{\infty }$ is a uniformly convergent sequence of functions.

We will now look at a nice theorem known as Cauchy's uniform convergence criterion for series of functions.

Theorem 1: Let $(f_{n}(x))_{n=1}^{\infty }$ be a sequence of real-valued functions with common domain $X$. Then $\displaystyle {\sum _{n=1}^{\infty }f_{n}(x)}$ is uniformly convergent on $X$ if and only if for all $\varepsilon >0$ there exists an $N\in \mathbb {N}$ such that if $n\geq N$ and for all $x\in X$ we have that $\displaystyle {\left|\sum _{k=n+1}^{n+p}f_{k}(x)\right|<\varepsilon }$ for all $p\in \mathbb {N}$.

• Proof: $\Rightarrow$ Suppose that $\displaystyle {\sum _{n=1}^{\infty }f_{n}(x)}$ is uniformly convergent to some limit function $f(x)$ on $X$. Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s_n(x))_{n=1}^{\infty}} denote the sequence of partial sums for this series. Then we must have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\lim_{n \to \infty} s_n(x) = f(x)}} uniformly on $X$. So, for $\varepsilon _{1}={\frac {\varepsilon }{2}}$ there exists an $N\in \mathbb {N}$ such that if $n\geq N$ and for all $x\in X$ we have that:

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

• For any $p\in \mathbb {N}$ let $m=n+p$. Then $m\geq N$ and so:

${\begin{aligned}\quad \quad \left|\sum _{k=1}^{m}f_{k}(x)-\sum _{k=1}^{n}f_{k}(x)\right|=\left|\sum _{k=1}^{n+p}f_{k}(x)-\sum _{k=1}^{n}f_{k}(x)\right|=\left|\sum _{k=n+1}^{n+p}f_{k}(x)\right|=\left|s_{m}(x)-s_{n}(x)\right|\leq \left|s_{m}(x)-f(x)\right|+\left|f(x)-s_{n}(x)\right|<\varepsilon _{1}+\varepsilon _{1}={\frac {\varepsilon }{2}}+{\frac {\varepsilon }{2}}=\varepsilon \end{aligned}}$

• $\Leftarrow$ Suppose that for all $\varepsilon >0$ there exists an $N\in \mathbb {N}$ such that if $n\geq N$ and for all $x\in X$ we have that:

${\begin{aligned}\quad \left|\sum _{k=n+1}^{n+p}f_{k}(x)\right|<\varepsilon \end{aligned}}$

• Let $m,n\geq N$. Assume without loss of generality that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m > n} and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = n + p} for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in \mathbb{N}} . Then from above we see that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \left| s_m(x) - s_n(x) \right| = \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x) \right| = \left| \sum_{k=n+1}^{n+p} f_k(x) \right| < \varepsilon \end{align}}

• So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s_n(x))_{n=1}^{\infty}} converges uniformly by the Cauchy uniform convergence criterion for sequences of functions. So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} converges uniformly on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

The Weierstrass M-Test for Uniform Convergence of Series of Functions

Recall that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} is a sequence of real-valued functions with common domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , then we say that the corresponding series of functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} is uniformly convergent if the sequence of partial sums Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s_n(x))_{n=1}^{\infty}} is a uniformly convergent sequence.

We will now look at a very nice and relatively simply test to determine uniform convergence of a series of real-valued functions called the Weierstrass M-test.

Theorem 1: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f_n(x))_{n=1}^{\infty}} be a sequence of real-valued functions with common domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M_n)_{n=1}^{\infty}} be a sequence of nonnegative real numbers such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| f_n(x) \right| \leq M_n} for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} and for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty} M_n}} converges then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} uniformly converges on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .

• Proof: Suppose that there exists a sequence of nonnegative real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (M_n)_{n=1}^{\infty}} such that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} and for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} we have that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \left| f_n(x) \right| \leq M_n \end{align}}

• Furthermore, suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty} M_n}} converges to some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \in \mathbb{R}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \geq 0} . Then we have that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \left| \sum_{n=1}^{\infty} f_n(x) \right| \leq \sum_{n=1}^{\infty} \left| f_n(x) \right| \leq \sum_{n=1}^{\infty} M_n = M \end{align}}

• So the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} converges for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in X} by the comparison test. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

Licensing

Content obtained and/or adapted from:

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