Series of Functions Review

Table of Contents

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

On the Pointwise Convergent and Uniformly Convergent Series of Functions page we said that a series of functions $\displaystyle{\sum_{n=1}^{\infty} f_n(x)}$ is Pointwise Convergent to a sum function $$f(x)$$ if the corresponding sequence of partial sums $(s_n(x))_{n=1}^{\infty}$ is pointwise convergent to $$f(x)$$ .

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 $$f(x)$$ .

(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 $\displaystyle{f'(c) = \sum_{n=1}^{\infty} f_n'(c)}$ .

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