Series of Functions Review

# Series of Functions Review

We will now review some of the recent material regarding series of functions.

• We also said that $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ is Uniformly Convergent to a sum function $f(x)$ if the corresponding sequence of partial sums $(s_n(x))_{n=1}^{\infty}$ is uniformly convergent to $f(x)$.
• In both cases, the general term for the sequence of partial sums is:
(1)
\begin{align} \quad s_n(x) = \sum_{k=1}^{n} f_n(x) = f_1(x) + f_2(x) + ... + f_n(x) \end{align}
• On the Cauchy's Uniform Convergence Criterion for Series of Functions we looked at a nice theorem which will tell us if a series of functions is uniformly convergent. We saw that $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ is uniformly convergent to some function $f(x)$ if and only if for all $\epsilon > 0$ there exists an $N \in \mathbb{N}$ such that if $n \geq N$ we have that for all $x \in X$ and for all $p \in \mathbb{N}$ that:
(2)
\begin{align} \quad \biggr \lvert \sum_{k=n+1}^{n+p} f_k(x) \biggr \rvert < \epsilon \end{align}
• We then looked at a specific test for uniform convergent. On The Weierstrass M-Test for Uniform Convergence of Series of Functions page. We saw that if $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ is a series of functions and if $(M_n)_{m=1}^{\infty}$ is a sequence of nonnegative real numbers such that $\mid f_n(x) \mid \leq M_n$ for all $x \in X$ and all $n \in \mathbb{N}$ and if $\displaystyle{\sum_{n=1}^{\infty} M_n}$ converges then $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges.
• We then looked at a bunch of properties that were preserved for series of functions. They are summarized below.
Page Hypotheses Conclusion
Continuity of the Sum of a Uniformly Convergent Series of Functions a) $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges to $f(x)$.
b) $f_n$ is continuous at a point $c \in X$ for all $n \in \mathbb{N}$.
$f$ is continuous at $c \in X$.
Riemann Integrability of the Sum of a Uniformly Convergent Series of Functions a) $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges to $f(x)$ on $[a, b]$.
b) $f_n$ is Riemann integrable on $[a, b]$ for all $n \in \mathbb{N}$.
$f$ is Riemann integrable on $[a, b]$ and $\displaystyle{\int_a^b f(x) \: dx = \int_a^b \sum_{n=1}^{\infty} f_n(x) \: dx = \sum_{n=1}^{\infty} \int_a^b f_n(x) \: dx}$.
Differentiability of the Sum of a Uniformly Convergent Series of Functions a) $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ uniformly converges to $f(x)$.
b) $f_n$ is differentiable at a point $c \in X$ for all $n \in \mathbb{N}$.
$f$ is differentiable at $c \in X$ and $\displaystyle{f'(c) = \sum_{n=1}^{\infty} f_n'(c)}$.