Friday, April 12, 2013

Series of functions

Definition: if $(f_n)$ is a sequence of functions defined on $D \subseteq \mathbb R$, $f_n : D \to \mathbb R$, the sequence of partial sums $(S_n)$ of the infinite series is defined for $x \in D$ as

$$ \begin{align*} s_1(x) &= f_1(x) \\ s_2(x) &= f_1(x) + f_2(x) \\ &\ldots \\ s_n(x) &= f_1(x) + f_2(x) + \cdots + f_n(x) \end{align*} $$

if $(s_n)$ converges on $D$ to a function $f$, we say $\sum f_n$ converges to $f$.

Definition: if $\sum |f_n(x)|$ converges for every $x \in D$ we say $\sum f_n$ converges absolutely on $D$.

Theorem: if $(f_n)$ are continuous on $D \subseteq \mathbb R$ to $\mathbb R$ $\forall n \in \mathbb N$ and $\sum f_n$ converges uniformly to $f$ on $D$ then $f$ is continuous on $D$.

Theorem: Let $(f_n)$ be a sequence of Riemann integrable functions on $J = [a,b]$. If $\sum f_n$ converges uniformly to $f$ on $J$ then $f$ is Riemann integrable on $J$ and $\int_a^b f = \sum\int_a^b f_n$. (Note that $\int_a^b f = \lim\int_a^b s_n$).

Theorem: Let $J = [a,b]$, $f_n : J \to \mathbb R$ be such that $f_n'$ exists on $J$ for all $n \in \mathbb N$. Suppose $\sum f_n$ converges for at least one point in $J$ and $\sum f_n'$ converges uniformly on $J$ then there exists an $f : J \to \mathbb R$ such that $\sum f_n \rightrightarrows f$ on $J$, $f$ is differentiable on $J$, and $f' = \sum f_n'$.

Given a series of functions $\sum f_n$ we have immediately two types of convergence, namely absolute and uniform. Does one imply the other? The answer is no, but think about it.

Cauchy Criterion: Let $(f_n)$ be a sequence of functions $f_n : D \to \mathbb R$ where $D \subseteq \mathbb R$. $\sum f_n$ converges uniformly on $D$ $\iff$ $\epsilon > 0$ given, there exists and $N \in \mathbb N$ fucn that if $m > n \geq N$, $|f_{n+1}(x) + f_{n+2} + \ldots + f_m(x)| < \epsilon$ for all $x \in D$.

Weierstrass M-test: Let $(M_n)$ be a sequence of positive real numbers such that $|f_n(x)| < M_n$ for all $x \in D$, where $f_n : D \to \mathbb R$. If $\sum M_n$ converges then the series $f_n$ converges uniformly on $D$.

Proof: Since the series $\sum M_n$ converges, for a given $\epsilon > 0$ there exists $N \in \mathbb N$ such that $if m > n \geq N$,

$$ |M_{n+1} + M_{n+2} + \ldots + M_m| = M_{n+1} + M_{n+2} + \ldots + M_m < \epsilon $$

thus if $m > n > N$, for any $x \in D$,

$$ |f_{n+1} + f_{n+2} + \ldots + f_m| \leq |f_{n+1}| + |f_{n+2}| + \ldots + |f_m| \leq M_{n+1} + M_{n+2} + \ldots + M_m < \epsilon $$

so $\sum f_n$ converges uniformly on $D$ by the Cauchy criterion.