8. Series of Functions

Point-wise Convergence

f_{n}, f : A \to R, \sum_{n = 1}^{\infty} f_{n} = f_{1} + f_{2} + f_{3} + \dots

converges point-wise if $S_{k} (x) = f_{1} (x) + \dots + f_{k} (x)$ converges point-wise

\sum f \overset{unif}{\to} f if S_{k} \overset{unif}{\to} f

Thm Term-by-term Continuity

If $f_{n}$ are cts on $A \subseteq R$
If $\sum_{}^{} f_{n} \overset{unif}{\to} f$ on $A$ then $f$ is continuous on $A$ by sequence theorem

Thm Term-by-term Differentiability

$f_{n} : I \to R$ and is differentiable and $\sum_{}^{} f_{n}^{'} \overset{unif}{\to} g$ on $I$ .
If there exists $x_{0} \in I$ with $\sum_{}^{} f_{n} (x_{0})$ coverages, then $\sum_{}^{} f_{n} \overset{unif}{\to} f$ where $f$ is differentiable and $f^{'} = g$

f (x) = \sum_{n = 1}^{\infty} f_{n} (x) a n d f (x) = \sum_{n = 1}^{\infty} f_{n}^{'} (x) = g (x) .

Thm Cauchy

$\sum_{}^{} f_{n}$ converges uniformly on $A \subseteq R$ iff $\forall ε > 0 \exists N \in N$ s.t

| f_{m + 1} (x) + \dots + f_{n} (x) | < ε

whenever $n > m \geq N$ and $x \in A$

Thm Weierstass M-Test

For each $n \in N$ let $f_{n} : A \to R$ and $M_{n} > 0$ with $| f_{n} (x) | \leq M_{n}$ for all $x \in A$

If $\sum_{}^{} M_{n}$ converges then $\sum_{}^{} f_{n}$ converges uniformly on $A$

Example:

\sum_{n = 0}^{\infty} \frac{1}{n!} x^{n} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots

definition of the exponential function

e^{x} = \sum_{}^{} \frac{1}{n!} x^{n}

converges for all $x \in R$ which is continuous everywhere and infinitely differentiable