$\newcommand{\br}{\\}$
$\newcommand{\R}{\mathbb{R}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\N}{\mathbb{N}}$
$\newcommand{\C}{\mathbb{C}}$
$\newcommand{\P}{\mathbb{P}}$
$\newcommand{\F}{\mathbb{F}}$
$\newcommand{\L}{\mathcal{L}}$
$\newcommand{\spa}[1]{\text{span}(#1)}$
$\newcommand{\dist}[1]{\text{dist}(#1)}$
$\newcommand{\max}[1]{\text{max}(#1)}$
$\newcommand{\min}[1]{\text{min}(#1)}$
$\newcommand{\supr}[1]{\text{sup}(#1)}$
$\newcommand{\infi}[1]{\text{inf}(#1)}$
$\newcommand{\set}[1]{\{#1\}}$
$\newcommand{\emptyset}{\varnothing}$
$\newcommand{\otherwise}{\text{ otherwise }}$
$\newcommand{\if}{\text{ if }}$
$\newcommand{\proj}{\text{proj}}$
$\newcommand{\union}{\cup}$
$\newcommand{\intercept}{\cap}$
$\newcommand{\abs}[1]{| #1 |}$
$\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$
$\newcommand{\pare}[1]{\left(#1\right)}$
$\newcommand{\t}[1]{\text{ #1 }}$
$\newcommand{\head}{\text H}$
$\newcommand{\tail}{\text T}$
$\newcommand{\d}{\text d}$
$\newcommand{\limu}[2]{\underset{#1 \to #2}\lim}$
$\newcommand{\inv}[1]{{#1}^{-1}}$
$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$
$\newcommand{\nullity}[1]{\text{nullity}(#1)}$
$\newcommand{\rank}[1]{\text{rank }#1}$
$\newcommand{\var}[1]{\text{var}(#1)}$
$\newcommand{\tr}[1]{\text{tr}(#1)}$
$\newcommand{\oto}{\text{ one-to-one }}$
$\newcommand{\ot}{\text{ onto }}$
$\newcommand{\Re}[1]{\text{Re}(#1)}$
$\newcommand{\Im}[1]{\text{Im}(#1)}$
$\newcommand{\Vcw}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Vce}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Vcr}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$
$\newcommand{\Vct}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$
$\newcommand{\Vcy}[6]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{bmatrix}}$
$\newcommand{\Vcu}[7]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{bmatrix}}$
$\newcommand{\vcw}[2]{\begin{matrix} #1 \br #2 \end{matrix}}$
$\newcommand{\vce}[3]{\begin{matrix} #1 \br #2 \br #3 \end{matrix}}$
$\newcommand{\vcr}[4]{\begin{matrix} #1 \br #2 \br #3 \br #4 \end{matrix}}$
$\newcommand{\vct}[5]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \end{matrix}}$
$\newcommand{\vcy}[6]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \end{matrix}}$
$\newcommand{\vcu}[7]{\begin{matrix} #1 \br #2 \br #3 \br #4 \br #5 \br #6 \br #7 \end{matrix}}$
$\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$
$\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$
$\newcommand{\Mqr}[4]{\begin{bmatrix} #1 & #2 & #3 & #4 \end{bmatrix}}$
$\newcommand{\Mqt}[5]{\begin{bmatrix} #1 & #2 & #3 & #4 & #5 \end{bmatrix}}$
$\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Mrq}[4]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \end{bmatrix}}$
$\newcommand{\Mtq}[5]{\begin{bmatrix} #1 \br #2 \br #3 \br #4 \br #5 \end{bmatrix}}$
$\newcommand{\Mqw}[2]{\begin{bmatrix} #1 & #2 \end{bmatrix}}$
$\newcommand{\Mwq}[2]{\begin{bmatrix} #1 \br #2 \end{bmatrix}}$
$\newcommand{\Mww}[4]{\begin{bmatrix} #1 & #2 \br #3 & #4 \end{bmatrix}}$
$\newcommand{\Mqe}[3]{\begin{bmatrix} #1 & #2 & #3 \end{bmatrix}}$
$\newcommand{\Meq}[3]{\begin{bmatrix} #1 \br #2 \br #3 \end{bmatrix}}$
$\newcommand{\Mwe}[6]{\begin{bmatrix} #1 & #2 & #3\br #4 & #5 & #6 \end{bmatrix}}$
$\newcommand{\Mew}[6]{\begin{bmatrix} #1 & #2 \br #3 & #4 \br #5 & #6 \end{bmatrix}}$
$\newcommand{\Mee}[9]{\begin{bmatrix} #1 & #2 & #3 \br #4 & #5 & #6 \br #7 & #8 & #9 \end{bmatrix}}$
Example
We show $\sum\ \frac{ 1 }{ n } = +\infty$.
Proof
</span>
</span>
<span class="proof__expand"><a>[expand]</a></span>
Example
We show $\sum \frac{ 1 }{ n^2 }$ converges.
Proof
</span>
</span>
<span class="proof__expand"><a>[expand]</a></span>
Definition: Alternating Series
Let $a_n >0 $, $\sum_{ n = 1 }^{ \infty } (-1)^{n+1} a_n$ is alternating.
any convergent series has $a_n \to 0 $.
Suppose $a_n > 0 $ and $a_n \to 0 $
suppose $a_n > a_{n+1} \forall n$ i.e. decreasing.
Theorem
$\sum \frac{ 1 }{ n^p } $ converges if and only if $p > 1 $.
Definition: Alternating Series
$\sum (-1)^{n+1} a_n $ is called an alternating series.
Theorem
: Alternating Series Theorem
If $a_1 \geq a_2 \geq … \geq a_n \geq … \geq 0 $ and $\lim a_n = 0 $, then the alternating series $\sum\ (-1)^{n +1} a_n $ converges. Moreover, the partial sums $s_n = \sum_{ k=1 }^{ n } (-1)^{k+1} a_k $ satisfy $\abs{ s - s_n } \leq a_n $for all $n $.
Proof
</span>
</span>
<span class="proof__expand"><a>[expand]</a></span>