📂Analysis

Functions of Series

Definitions

Let’s define the series of functions $\left\{ f_{n} : E \to \mathbb{R} \right\}_{n=1}^{\infty}$.

(1) If $\displaystyle \sum_{k=1}^{n} f_{k} (X)$ when $n \to \infty$, then the series $\displaystyle \sum_{k=1}^{ \infty } f_{k}$ is said to converge pointwise in $E$ if it converges pointwise.

(2) If $\displaystyle \sum_{k=1}^{n} f_{k} (X)$ when $n \to \infty$, then the series $\displaystyle \sum_{k=1}^{ \infty } f_{k}$ is said to converge uniformly in $E$ if it converges uniformly.

(3) If $\displaystyle \sum_{k=1}^{n} | f_{k} (x) |$ when $n \to \infty$, then the series $\displaystyle \sum_{k=1}^{ \infty } f_{k}$ is said to converge absolutely in $E$ if it converges pointwise.

Explanation

Having discussed sequences of functions, it is indispensable to talk about series. Unlike mere convergence of function sequences, here we also consider absolute convergence.

Theorem

Assume $\displaystyle F := \sum_{k=1}^{ \infty } f_{k}$ converges uniformly in $E$.

(1) Continuity: If $f_{n}$ is continuous at $x_{0} \in E$, then $F$ is also continuous at $x_{0} \in E$.

(2) Differentiability: If $f_{n}$ is differentiable at $E = (a,b)$ and $\displaystyle \sum_{k=1}^{\infty} f_{n} '$ converges uniformly in $E$, then $\displaystyle F$ is also differentiable in $E$ and

$${{ d } \over { dx }} \sum_{k=1}^{\infty} f_{n} (x) = \sum_{k=1}^{\infty} {{ d } \over { dx }} f_{n} (x)$$

(3) Integrability: If $f_{n}$ is integrable in $E = [a,b]$, then $F$ is also integrable in $E$ and

$$\lim_{n \to \infty} \int_{a}^{b} f_{n} (x) dx = \int_{a}^{b} \left( \lim_{n \to \infty} f_{n} (x) \right) dx$$

(4) Weierstrass M-test:

Given a sequence of functions $\left\{ f_{n} \right\}$ and $x \in E$, if there exists a sequence of positive numbers $M_{n}$ satisfying $|f_{n}(z)| \le M_{n}$ and $\displaystyle \sum_{n=1}^{\infty} M_{n}$ converges, then $\displaystyle \sum_{n=1}^{\infty} f_{n}$ converges absolutely and uniformly in $E$.

(5) Dirichlet’s test:

For sequences of functions $\left\{ f_{k} \right\}$, $\left\{ g_{k} \right\}$ and $n \in \mathbb{N}$, $x \in E$, if there exists a positive number $M$ satisfying $\displaystyle \left| \sum_{k=1}^{n} f_{k} (x) \right| \le M < \infty$ and $g_{k}$ converges uniformly to $g = 0$ in $E$, then $\displaystyle \sum_{k=1}^{\infty} f_{k} g_{k}$ also converges uniformly in $E$.