Uniform Convergence and Power Series

Definition

A \(\underline{\textbf{power series}}\) of \(f(x)\) is the infinite sum \(\lim_{n\to\infty}T_{(n,f,a)}\).

Introduction

\[\text{The power series of } e^x \\ = \lim_{n\to\infty} T_{n,0} = 1+x+\frac{x^2}{2!} +\ldots+\frac{x^n}{n!}+\frac{x^{n+1}}{(n+1)!}\\\\\] \[\text{Interested in functions of the form:} \\ f(x) = f_1(x) + f_2(x) + f_3(x) + \ldots \\\\ \text{In other words, we have a sequence of sums:} \\ \{f_1(x), f_1(x) + f_2(x), f_1(x) + f_2(x) + f_3(x), \ldots\} \\ s_n = f_1 + \ldots + f_n \\ \{s_n\} = \{s_1,s_2,s_3,\ldots\} \\ f(x) = \lim_{n\to\infty} f_n(x) \\\\ \text{For such functions, $\lim_{n\to\infty} f_n(x)$,} \\ \text{nothing we hope to be true, is true, for example:} \\\]

1. \[f_n(x) \text{ continuous } \nRightarrow \lim_{n\to\infty} f_n(x) \text{ continuous }\]
2. \[\lim_{n\to\infty} \int_0^1 f_n(x)dx \nRightarrow \int_0^1 \lim_{n\to\infty} f_n(x)dx\]
3. \[f_n(x) \text{ differentiable } \nRightarrow \lim_{n\to\infty} f_n(x) \text{ differentiable }\]

\[\text{Some counter-examples:} \\\\ \text{Counter-example for continuity:} \\ 0 \leq x \leq 1,\ f_n(x) = x^n \\ \implies f_n(x) \text{ continuous } \\ \lim_{n\to\infty} f_n(x) = f(x) = \begin{cases} 0, & \text{ if } 0 \leq x < 1 \\ 1, & \text{ if } x = 1 \end{cases} \\ \implies \lim_{n\to\infty} f_n(x) \text{ not continuous } \\\\ \text{Counter-example for integrals:} \\ 0 \leq x \leq 1, \\ f_n(x) \text{ defined as line connecting $(\frac{1}{2n},2n)$ to $(\frac 1 n, 0)$ to $(1,0)$} \\ \implies \int_0^1 f_n(x) = \frac 1 2 \cdot \frac 1 n \cdot 2n = 1 \\ \lim_{n\to\infty} f_n(x) = f(x) = 0 \\ \implies \int_0^1 \lim_{n\to\infty} f_n(x) = 0 \\ \implies \lim_{n\to\infty} \int_0^1 f_n(x) \neq \int_0^1 \lim_{n\to\infty} f_n(x) \\\\ \text{Counter-example for differentiability:} \\ x\in [-1,1],\ f_n(x) = \sqrt{x^2+\frac 1 n} \\ \implies f_n(x) \text{ differentiable } \\ \lim_{n\to\infty} f_n(x) = \left|x\right| \\ \implies \lim_{n\to\infty} f_n(x) \text{ not differentiable } \\\\\]

Definitions

Let \(\{f_n(x)\}\) be a sequence of functions defined on the set \(A\) (domain), and let \(f(x)\) be a function which is also defined \(\forall\ x\in A\). Then \(f(x)\) is called the \(\underline{\textbf{uniform limit of } \{f_n(x)\} \textbf{ on } A}\) if:

\[\forall\ \epsilon > 0, \exists N, \text{ s.t. } \forall\ x\in A, n>N \implies \left|f(x) - f_n(x)\right| < \epsilon\]

We also say that \(\underline{\{f_n(x)\} \textbf{ converges uniformly to } f \textbf{ on } A}\) or \(\underline{f_n(x) \textbf{ approaches } f(x) \textbf{ uniformly on } A}\).

\(\underline{\{f_n(x)\} \textbf{ converges pointwise to } f \textbf{ on } A}\) if:

\[\forall\ x, f(x) = \lim_{n\to\infty} f_n(x) \\ \forall\ \epsilon > 0, \forall\ x\in A, \exists N, \text{ s.t. } n>N \implies \left|f(x) - f_n(x)\right| < \epsilon\]

Note) \(\text{uniform convergence }\implies\text{ pointwise convergence }\), but \(\text{ pointwise convergence }\nRightarrow\text{ uniform convergence }\) because we cannot necessarily find \(N\) that works \(\forall\ x\)

Note) \(\text{uniform convergence }\implies \text{ after } n>N\), all functions will be between \(f(x)-\epsilon\) and \(f(x)+\epsilon\)

Applications 1

• 1.) If \(f_n(x) = \frac x n \text{ on } [0,1]\), what does \(\{f_n(x)\}\) approach uniformly?
  • \[\lim_{n\to\infty} f_n(x) = 0 \\ \implies \{f_n(x)\} \text{ approaches } f(x) = 0 \text{ uniformly on } [0,1]\]
  • Answer: \(\{f_n(x)\} \text{ approaches } f(x) = 0 \text{ uniformly on } [0,1]\)
• 2.) Let \(0<a<1\), Prove \(\{x^n\}\) approaches \(0\) uniformly on the interval \([0,a]\).
  • \[\begin{align} \text{WWTP: } \forall\ \epsilon > 0, \exists N, \text{ s.t. } \forall\ x\in [0,a], n>N &\implies \left|f(x) - f_n(x)\right| < \epsilon \\ &\implies |0 - f_n(x)| < \epsilon \\ &\implies |x^n| < \epsilon \\\\ \end{align}\]
  • \[\text{We need to find N large enough that } a^N < \epsilon \\ \text{Performing preliminary "scratch-work":}\]
  • \[\begin{align} 0<a<1 &\implies \log_a \text{ is a decreasing function } \\ &\implies \log_a a^N > \log_a \epsilon \\ &\implies N > \log_a \epsilon \\\\ \end{align}\]
  • \[\text{Choose $N$ s.t. $N>\log_a \epsilon$:}\]
  • \[\begin{align} &\implies \log_a a^N > \log_a \epsilon \\ &\implies a^N < \epsilon \\\\ \end{align}\]
  • \[\text{Assume $n>N$:}\]
  • \[\begin{align} &\implies a^n < a^N < \epsilon \\ &\implies x^n \leq a^n < a^N < \epsilon &&(\forall\ x \in [0,a], x\leq a) \\ &\implies x^n < \epsilon \\ &\implies |x^n| < \epsilon &&(x\geq0) \ \ \ \ \blacksquare\\\\ \end{align}\]

Theorem 1

\[\begin{align} &(\{f_n(x)\} \text{ integrable on } [a,b]) \\ &\land(f(x) \text{ integrable on } [a,b]) \\ &\land(\{f_n(x)\} \text{ converges uniformly to } f(x) \text{ on } [a,b]) \\ &\implies \int_a^b f(x)dx = \lim_{n\to\infty} \int_a^b f_n(x)dx \end{align}\]

Proof.

Refer to spivak pg 504, rough notes pg 8.

Theorem 2

\[\begin{align} &(\{f_n(x)\} \text{ converges uniformly to } f(x) \text{ on } [a,b]) \\ &\land(\forall\ n, f_n(x) \text{ continuous on } [a,b]) \\ &\implies f(x) \text{ continuous on } [a,b] \end{align}\]

Proof.

Refer to spivak pg 503, rough notes pg 8.

Theorem 3

\[\begin{align} &(\forall\ n, f_n(x) \text{ differentiable on } [a,b]) \\ &\land(\forall\ n, f_n(x) \text{ has integrable derivatives } f_n'(x)) \\ &\land(\{f_n(x)\} \text{ converges pointwise to } f(x) \text{ on } [a,b]) \\ &\land(g(x) \text{ continuous }) \\ &\land(\{f_n'(x)\} \text{ converges uniformly to } g(x) \text{ on } [a,b]) \\ &\implies (f(x) \text{ differentiable on } [a,b]) \\ &\phantom{00000}\land (f'(x) = \lim_{n\to\infty} f_n'(x)) \end{align}\]

Proof.

Refer to spivak pg 505.

Definition

The series \(\sum_{n=1}^\infty f_n(x)\) \(\underline{\textbf{converges uniformly}}\) (more formally : the sequence \(\{f_n\}\) is \(\underline{\textbf{uniformly summable}}\)) \(\underline{\textbf{to } f \textbf{ on } A}\), if the sequence of partial sums \(\{f_1,f_1+f_2,\ldots\}\) converges uniformly to \(f\) on \(A\).

Corollary

\[(\sum_{n=1}^\infty f_n(x) \text{ converges uniformly to } f(x) \text{ on } [a,b])\] \[\begin{align} 1.) &\land(\forall\ n, f_n(x) \text{ continuous } on [a,b]) \\ &\implies f(x) \text{ continuous on } [a,b]\\\\ 2.) &\land(f(x) \text{ integrable on } [a,b] ) \\ &\land(f_n(x) \text{ integrable on } [a,b] ) \\ &\implies \int_a^b f(x)dx = \sum_{n=1}^\infty \int_a^b f_n(x)dx\\\\ 3.)&\land(\sum_{n=1}^\infty f_n(x) \text{ converges pointwise to } f(x) \text{ on } [a,b]) \\ &\land(\forall\ n, f_n(x) \text{ has integrable derivative } f_n'(x)) \\ &\land(\sum_{n=1}^\infty f_n'(x) \text{ converges uniformly to } g(x) \text{ on } [a,b]) \\ &\land(g(x) \text{ continuous } ) \\ &\implies \forall\ x \in [a,b], f'(x) = \sum_{n=1}^\infty f_n'(x) \end{align}\]

Definitions

A \(\underline{\textbf{power series}}\) has the form:

\[\sum_{n=0}^\infty b_n(x-a)^n \\ (\forall\ n, b_n \text{ is a constant})\]

The \(\underline{\textbf{Taylor Series}}\) for \(f(x)\) at \(x=a\) is:

\[\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n\]

Weierstrass M-Test Theorem

\[\begin{align} &(\forall\ f_n(x) \in \{f_n(x)\}, f_n(x) \text{ defined on } A) \\ &\land(\forall\ x \in A, \exists\ \{M_n\} \text{ s.t. } \forall\ n, |f_n(x)| \leq M_n, M_n \text{ is a constant }) \\ &\land (\sum_{n=1}^\infty M_n \text{ converges }) \\ &\implies (\forall x \in A, \sum_{n=1}^\infty f_n(x) \text{ converges absolutely})\\ &\phantom{00000}\land (\forall x \in A, \sum_{n=1}^\infty f_n(x) \text{ converges }) \\ &\phantom{00000}\land (\sum_{n=1}^\infty f_n(x) \text{ converges uniformly to } f(x)=\sum_{n=1}^\infty f_n(x) \text{ on } A) \end{align}\]

Proof.

Refer to spivak pg 508, rough notes pg 9.

Theorem 4

\[\begin{align} &(f(x_0)=\sum_{n=0}^\infty a_nx_0^n \text{ converges }) \\ &\land(\exists\ a \in (0,|x_0|)) \\ &\implies(f(x)=\sum_{n=0}^\infty a_nx^n \text{ converges uniformly (and absolutely) on } [-a,a]) \\ &\phantom{00000}\land(g(x)=\sum_{n=0}^\infty na_nx^{n-1} \text{ converges uniformly (and absolutely) on } [-a,a]) \\ &\phantom{00000}\land(f \text{ differentiable }) \\ &\phantom{00000}\land(\forall\ |x| < |x_0|, f'(x)=\sum_{n=1}^\infty na_nx^{n-1}) \end{align}\]

Proof.

Refer to spivak pg 512, rough notes pg 9.

Definitions

\[\begin{align} &\exists\ R \text{ (called the } \underline{\textbf{radius of convergence}} \text{) s.t.}\\ &(f(x)=\sum_{n=0}^\infty a_nx^n) \\ &\land(\exists\ R \geq 0 \text{ or } R = \infty) \\ &1.)\land(\forall\ |x| < R) \implies f(x) \text{ converges absolutely } \\ &2.)\land(\forall\ |x| > R) \implies f(x) \text{ diverges} \\\\ \end{align}\]

The interval on which \(f(x)\) converges absolutely is called the \(\underline{\textbf{interval of convergence}}\).