Series of Functions Review

# Series of Functions Review

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 $f(x)$.

- In both cases, the general term for the sequence of partial sums is:

\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:

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