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# pkra/mse-tutorial-plain.html

Last active July 4, 2018 07:15
TeX sample extracted from math.SE tutorial
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 Math.SE extracted Samples

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$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$
$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$
$$\alpha, \beta, … \omega$$
$$\Gamma, \Delta, …, \Omega$$
$$x_i^2$$
$$\log_2 x$$
$$10^10$$
$$10^{10}$$
$${x^y}^z$$
$$x^{y^z}$$
$$x_i^2$$
$$x_{i^2}$$
$$(2+3)[4+4]$$
$$\{\}$$
$$(\frac{\sqrt x}{y^3})$$
$$\left(\frac{\sqrt x}{y^3}\right)$$
$$(x)$$
$$[x]$$
$$\{ x \}$$
$$|x|$$
$$\vert x \vert$$
$$\Vert x \Vert$$
$$\langle x \rangle$$
$$\lceil x \rceil$$
$$\lfloor x \rfloor$$
$$\left.\frac12\right\rbrace$$
$$\Biggl(\biggl(\Bigl(\bigl((x)\bigr)\Bigr)\biggr)\Biggr)$$
$$\sum_1^n$$
$$\sum_{i=0}^\infty i^2$$
$$\prod$$
$$\int$$
$$\bigcup$$
$$\bigcap$$
$$\iint$$
$$\iiint$$
$$\frac ab$$
$$\frac{a+1}{b+1}$$
$${a+1\over b+1}$$
$$\cfrac{a}{b}$$
$$\mathbb{CHNQRZ}$$
$$\mathbf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathbf{abcdefghijklmnopqrstuvwxyz}$$
$$\mathtt{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathtt{abcdefghijklmnopqrstuvwxyz}$$
$$\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathrm{abcdefghijklmnopqrstuvwxyz}$$
$$\mathsf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathsf{abcdefghijklmnopqrstuvwxyz}$$
$$\mathcal{ ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$$
$$\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \mathfrak{abcdefghijklmnopqrstuvwxyz}$$
$$\sqrt{x^3}$$
$$\sqrt[3]{\frac xy}$$
$$\sin x$$
$$sin x$$
$\lim_{x\to 0}$
$$\lt\, \gt\, \le\, \leq\, \leqq\, \leqslant\, \ge\, \geq\, \geqq\, \geqslant\, \neq$$
$$\not\lt$$
$$\times\, \div\, \pm\, \mp$$
$$x\cdot y$$
$$\cup\, \cap\, \setminus\, \subset\, \subseteq \,\subsetneq \,\supset\, \in\, \notin\, \emptyset\, \varnothing$$
$${n+1 \choose 2k}$$
$$\to\, \rightarrow\, \leftarrow\, \Rightarrow\, \Leftarrow\, \mapsto$$
$$\land\, \lor\, \lnot\, \forall\, \exists\, \top\, \bot\, \vdash\, \vDash$$
$$\star\, \ast\, \oplus\, \circ\, \bullet$$
$$\approx\, \sim \, \simeq\, \cong\, \equiv\, \prec, \lhd$$
$$\infty\, \aleph_0$$
$$\nabla\, \partial$$
$$\Im\, \Re$$
$$a\equiv b\pmod n$$
$$a_1, a_2, \ldots ,a_n$$
$$a_1+a_2+\cdots+a_n$$
$$\epsilon\, \varepsilon$$
$$\phi\, \varphi$$
$$\ell$$
$$\TeX$$
$$\LaTeX$$
$$\TeX$$
$$a b$$
$$a\,b$$
$$a\;b$$
$$a\quad b$$
$$a\qquad b$$
$$\{x\in s\mid x\text{ is extra large}\}$$
$$\hat x$$
$$\widehat{xy}$$
$$\bar x$$
$$\overline{xyz}$$
$$\vec x$$
$$\overrightarrow{xy}$$
$$\overleftrightarrow{xy}$$
$$\frac d{dx}x\dot x = \dot x^2 + x\ddot x$$
$$\$$
$$\{$$
$$\_$$
$$\backslash$$
$$\LaTeX$$
$$\mathrm{B}$$
$$\sin$$
$$\operatorname{Spec} A$$
$$_5C_3$$
$$\TeX$$
$$\TeX$$
$$\LaTeX$$
$$\mathrm{d}x$$
$\begin{matrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \\ \end{matrix}$
$$\begin{pmatrix}1&2\\3&4\\ \end{pmatrix}$$
$$\begin{bmatrix}1&2\\3&4\\ \end{bmatrix}$$
$$\begin{Bmatrix}1&2\\3&4\\ \end{Bmatrix}$$
$$\begin{vmatrix}1&2\\3&4\\ \end{vmatrix}$$
$$\begin{Vmatrix}1&2\\3&4\\ \end{Vmatrix}$$
$$\cdots$$
$$\ddots$$
$$\vdots$$
\begin{pmatrix} 1 & a_1 & a_1^2 & \cdots & a_1^n \\ 1 & a_2 & a_2^2 & \cdots & a_2^n \\ \vdots & \vdots& \vdots & \ddots & \vdots \\ 1 & a_m & a_m^2 & \cdots & a_m^n \end{pmatrix}
$\left[\begin{array}{cc|c} 1&2&3\\ 4&5&6 \end{array}\right]$
$\begin{pmatrix} a & b \\ c & d\\ \hline 1 & 0\\ 0 & 1 \end{pmatrix}$
$$\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)$$
\begin{align} \sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\ & = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ & = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\ & = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ & \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right) \end{align}
\begin{align} f(x)&=\left(x^3\right)+\left(x^3+x^2+x^1\right)+\left(x^3+x^2\right)\\ f'(x)&=\left(\left(3x^2+2x+1\right)+\left(3x^2+2x\right)\right)\\ f''(x)&=\left(6x+2\right)\\ \end{align}
$${}{}{}{}{}{}{}{}$$
$$\Psi$$
$$\delta$$
$$\zeta$$
$$\ge$$
$$\subseteq$$
$$\LaTeX$$
$$\LaTeX$$
f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases}
$\left. \begin{array}{l} \text{if n is even:}&n/2\\ \text{if n is odd:}&3n+1 \end{array} \right\} =f(n)$
f(n) = \begin{cases} \frac{n}{2}, & \text{if $n$ is even} \$2ex] 3n+1, & \text{if n is odd} \end{cases} \begin{array}{c|lcr} n & \text{Left} & \text{Center} & \text{Right} \\ \hline 1 & 0.24 & 1 & 125 \\ 2 & -1 & 189 & -8 \\ 3 & -20 & 2000 & 1+10i \end{array} \[ % outer vertical array of arrays \begin{array}{c} % inner horizontal array of arrays \begin{array}{cc} % inner array of minimum values \begin{array}{c|cccc} \text{min} & 0 & 1 & 2 & 3\\ \hline 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 1 & 1\\ 2 & 0 & 1 & 2 & 2\\ 3 & 0 & 1 & 2 & 3 \end{array} & % inner array of maximum values \begin{array}{c|cccc} \text{max}&0&1&2&3\\ \hline 0 & 0 & 1 & 2 & 3\\ 1 & 1 & 1 & 2 & 3\\ 2 & 2 & 2 & 2 & 3\\ 3 & 3 & 3 & 3 & 3 \end{array} \end{array} \\ % inner array of delta values \begin{array}{c|cccc} \Delta&0&1&2&3\\ \hline 0 & 0 & 1 & 2 & 3\\ 1 & 1 & 0 & 1 & 2\\ 2 & 2 & 1 & 0 & 1\\ 3 & 3 & 2 & 1 & 0 \end{array} \end{array}$
$$\mathsf{Show\ Math\ As\ }\blacktriangleright\mathsf{\ TeX\ Commands}$$
$$all italics, weird-looking spacing, an' odd apostrophes$$
$$\LaTeX$$
$\begin{array}{ll} \hfill\mathrm{Bad}\hfill & \hfill\mathrm{Better}\hfill \\ \hline \\ e^{i\frac{\pi}2} \quad e^{\frac{i\pi}2}& e^{i\pi/2} \\ \int_{-\frac\pi2}^\frac\pi2 \sin x\,dx & \int_{-\pi/2}^{\pi/2}\sin x\,dx \\ \end{array}$
\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ e^{i\frac{\pi}2} \quad e^{\frac{i\pi}2}& e^{i\pi/2} \\ \int_{-\frac\pi2}^\frac\pi2 \sin x\,dx & \int_{-\pi/2}^{\pi/2}\sin x\,dx \\ \end{array}
\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ \{x|x^2\in\Bbb Z\} & \{x\mid x^2\in\Bbb Z\} \\ \end{array}
\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ \int\int_S f(x)\,dy\,dx & \iint_S f(x)\,dy\,dx \\ \int\int\int_V f(x)\,dz\,dy\,dx & \iiint_V f(x)\,dz\,dy\,dx \end{array}
$$\TeX$$
\begin{array}{cc} \mathrm{Bad} & \mathrm{Better} \\ \hline \\ \iiint_V f(x)dz dy dx & \iiint_V f(x)\,dz\,dy\,dx \end{array}
$$\left\{x\middle | \frac{x^2}{2} \in \mathbb{z}\right\}$$
$\require{cancel}\begin{array}{rl} \verb|y+\cancel{x}| & y+\cancel{x}\\ \verb|\cancel{y+x}| & \cancel{y+x}\\ \verb|y+\bcancel{x}| & y+\bcancel{x}\\ \verb|y+\xcancel{x}| & y+\xcancel{x}\\ \verb|y+\cancelto{0}{x}| & y+\cancelto{0}{x}\\ \verb+\frac{1\cancel9}{\cancel95} = \frac15+& \frac{1\cancel9}{\cancel95} = \frac15 \\ \end{array}$
$\require{enclose}\begin{array}{rl} \verb|\enclose{horizontalstrike}{x+y}| & \enclose{horizontalstrike}{x+y}\\ \verb|\enclose{verticalstrike}{\frac xy}| & \enclose{verticalstrike}{\frac xy}\\ \verb|\enclose{updiagonalstrike}{x+y}| & \enclose{updiagonalstrike}{x+y}\\ \verb|\enclose{downdiagonalstrike}{x+y}| & \enclose{downdiagonalstrike}{x+y}\\ \verb|\enclose{horizontalstrike,updiagonalstrike}{x+y}| & \enclose{horizontalstrike,updiagonalstrike}{x+y}\\ \end{array}$
$$19/95 = 1/5$$
$$\cancelto{\cancelto{\cancelto{x^{2+x}}{\cancelto{x^2}{x}+4}}4}0$$
$$\LaTeX$$
$\left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1 \\ a_2x+b_2y+c_2z=d_2 \\ a_3x+b_3y+c_3z=d_3 \end{array} \right.$
$\begin{cases} a_1x+b_1y+c_1z=d_1 \\ a_2x+b_2y+c_2z=d_2 \\ a_3x+b_3y+c_3z=d_3 \end{cases}$
\left\{\begin{aligned} a_1x+b_1y+c_1z&=d_1+e_1 \\ a_2x+b_2y&=d_2 \\ a_3x+b_3y+c_3z&=d_3 \end{aligned} \right.
$\left\{\begin{array}{ll}a_1x+b_1y+c_1z &=d_1+e_1 \\ a_2x+b_2y &=d_2 \\ a_3x+b_3y+c_3z &=d_3 \end{array} \right.$
$\begin{cases} a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\[2ex] a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\[2ex] a_3x+b_3y+c_3z=\frac{p_3}{q_3} \end{cases}$
$\begin{cases} a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\ a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\ a_3x+b_3y+c_3z=\frac{p_3}{q_3} \end{cases}$
$\left\{ \begin{array}{l} 0 = c_x-a_{x0}-d_{x0}\dfrac{(c_x-a_{x0})\cdot d_{x0}}{\|d_{x0}\|^2} + c_x-a_{x1}-d_{x1}\dfrac{(c_x-a_{x1})\cdot d_{x1}}{\|d_{x1}\|^2} \\[2ex] 0 = c_y-a_{y0}-d_{y0}\dfrac{(c_y-a_{y0})\cdot d_{y0}}{\|d_{y0}\|^2} + c_y-a_{y1}-d_{y1}\dfrac{(c_y-a_{y1})\cdot d_{y1}}{\|d_{y1}\|^2} \end{array} \right.$
$$\longrightarrow$$
$\begin{array}{|rc|} \hline \verb+\color{black}{text}+ & \color{black}{text} \\ \verb+\color{gray}{text}+ & \color{gray}{text} \\ \verb+\color{silver}{text}+ & \color{silver}{text} \\ \verb+\color{white}{text}+ & \color{white}{text} \\ \hline \verb+\color{maroon}{text}+ & \color{maroon}{text} \\ \verb+\color{red}{text}+ & \color{red}{text} \\ \verb+\color{yellow}{text}+ & \color{yellow}{text} \\ \verb+\color{lime}{text}+ & \color{lime}{text} \\ \verb+\color{olive}{text}+ & \color{olive}{text} \\ \verb+\color{green}{text}+ & \color{green}{text} \\ \verb+\color{teal}{text}+ & \color{teal}{text} \\ \verb+\color{aqua}{text}+ & \color{aqua}{text} \\ \verb+\color{blue}{text}+ & \color{blue}{text} \\ \verb+\color{navy}{text}+ & \color{navy}{text} \\ \verb+\color{purple}{text}+ & \color{purple}{text} \\ \verb+\color{fuchsia}{text}+ & \color{magenta}{text} \\ \hline \end{array}$
$$r, g, b$$
$$0–15$$
$\begin{array}{|rrrrrrrr|}\hline \verb+#000+ & \color{#000}{text} & & & \verb+#00F+ & \color{#00F}{text} & & \\ & & \verb+#0F0+ & \color{#0F0}{text} & & & \verb+#0FF+ & \color{#0FF}{text}\\ \verb+#F00+ & \color{#F00}{text} & & & \verb+#F0F+ & \color{#F0F}{text} & & \\ & & \verb+#FF0+ & \color{#FF0}{text} & & & \verb+#FFF+ & \color{#FFF}{text}\\ \hline \end{array}$
\begin{array}{|rrrrrrrr|} \hline \verb+#000+ & \color{#000}{text} & \verb+#005+ & \color{#005}{text} & \verb+#00A+ & \color{#00A}{text} & \verb+#00F+ & \color{#00F}{text} \\ \verb+#500+ & \color{#500}{text} & \verb+#505+ & \color{#505}{text} & \verb+#50A+ & \color{#50A}{text} & \verb+#50F+ & \color{#50F}{text} \\ \verb+#A00+ & \color{#A00}{text} & \verb+#A05+ & \color{#A05}{text} & \verb+#A0A+ & \color{#A0A}{text} & \verb+#A0F+ & \color{#A0F}{text} \\ \verb+#F00+ & \color{#F00}{text} & \verb+#F05+ & \color{#F05}{text} & \verb+#F0A+ & \color{#F0A}{text} & \verb+#F0F+ & \color{#F0F}{text} \\ \hline \verb+#080+ & \color{#080}{text} & \verb+#085+ & \color{#085}{text} & \verb+#08A+ & \color{#08A}{text} & \verb+#08F+ & \color{#08F}{text} \\ \verb+#580+ & \color{#580}{text} & \verb+#585+ & \color{#585}{text} & \verb+#58A+ & \color{#58A}{text} & \verb+#58F+ & \color{#58F}{text} \\ \verb+#A80+ & \color{#A80}{text} & \verb+#A85+ & \color{#A85}{text} & \verb+#A8A+ & \color{#A8A}{text} & \verb+#A8F+ & \color{#A8F}{text} \\ \verb+#F80+ & \color{#F80}{text} & \verb+#F85+ & \color{#F85}{text} & \verb+#F8A+ & \color{#F8A}{text} & \verb+#F8F+ & \color{#F8F}{text} \\ \hline \verb+#0F0+ & \color{#0F0}{text} & \verb+#0F5+ & \color{#0F5}{text} & \verb+#0FA+ & \color{#0FA}{text} & \verb+#0FF+ & \color{#0FF}{text} \\ \verb+#5F0+ & \color{#5F0}{text} & \verb+#5F5+ & \color{#5F5}{text} & \verb+#5FA+ & \color{#5FA}{text} & \verb+#5FF+ & \color{#5FF}{text} \\ \verb+#AF0+ & \color{#AF0}{text} & \verb+#AF5+ & \color{#AF5}{text} & \verb+#AFA+ & \color{#AFA}{text} & \verb+#AFF+ & \color{#AFF}{text} \\ \verb+#FF0+ & \color{#FF0}{text} & \verb+#FF5+ & \color{#FF5}{text} & \verb+#FFA+ & \color{#FFA}{text} & \verb+#FFF+ & \color{#FFF}{text} \\ \hline \end{array}
$$\def\demo#1#2{#1{#2}\ #1{#2#2}\ #1{#2#2#2}}$$
$$\demo\overline A$$
$$\demo\underline B$$
$$\demo\widetilde C$$
$$\demo\widehat D$$
$$\demo\fbox {E}$$
$$\demo\underleftarrow{F}\qquad$$
$$\xleftarrow{abc}$$
$$\demo\underrightarrow{G}\qquad$$
$$\xrightarrow{abc}$$
$$\demo\underleftrightarrow{H}$$
$$\overbrace{(n - 2) + \overbrace{(n - 1) + n + (n + 1)} + (n + 2)}$$
$$(n \underbrace{- 2) + (n \underbrace{- 1) + n + (n +} 1) + (n +} 2)$$
$\underbrace{a\cdot a\cdots a}_{b\text{ times}}$
$$\varliminf$$
$$\varlimsup$$
$$\check{I}$$
$$\acute{J}$$
$$\grave{K}$$
$$\vec u\ \vec{AB}$$
$$\bar z$$
$$\hat x$$
$$\tilde x$$
$$\dot x,\ddot x,\dddot x$$
$$\mathring A$$
$$\overset{@}{ABC}\ \overset{x^2}{\longmapsto}\ \overset{\bullet\circ\circ\bullet}{T}$$
$x = a_0 + \cfrac{1^2}{a_1 + \cfrac{2^2}{a_2 + \cfrac{3^2}{a_3 + \cfrac{4^4}{a_4 + \cdots}}}}$
$x = a_0 + \frac{1^2}{a_1 + \frac{2^2}{a_2 + \frac{3^2}{a_3 + \frac{4^4}{a_4 + \cdots}}}}$
$x = a_0 + \frac{1^2}{a_1+} \frac{2^2}{a_2+} \frac{3^2}{a_3 +} \frac{4^4}{a_4 +} \cdots$
$$[a_0; a_1, a_2, a_3, \ldots]$$
$\cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\ddots }}}= {\genfrac{}{}{}{}{a_1}{b_1}} {\genfrac{}{}{0pt}{}{}{+}} {\genfrac{}{}{}{}{a_2}{b_2}} {\genfrac{}{}{0pt}{}{}{+}} {\genfrac{}{}{}{}{a_3}{b_3}} {\genfrac{}{}{0pt}{}{}{+\dots}}$
$$\LaTeX$$
$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$
$\mathop{\LARGE\mathrm K}_{i=1}^\infty \frac{a_i}{b_i}$
$\require{AMScd} \begin{CD} A @>a>> B\\ @V b V V= @VV c V\\ C @>>d> D \end{CD}$
$\begin{CD} A @>>> B @>{\text{very long label}}>> C \\ @. @AAA @| \\ D @= E @<<< F \end{CD}$
$\require{AMScd} \begin{CD} RCOHR'SO_3Na @>{\text{Hydrolysis,\Delta, Dil.HCl}}>> (RCOR')+NaCl+SO_2+ H_2O \end{CD}$
$$\newcommand{\SES}[3]{ 0 \to #1 \to #2 \to #3 \to 0 }$$
$\SES{A}{B}{C}$
$$\implies$$
$$\Rightarrow$$
$$\iff$$
$$\impliedby$$
$$\to$$
$$f\colon A \to B$$
$$\gets$$
$$T:\mathbb R\to \mathbb R,\; x\mapsto x+1$$
$$p\land((q\lor r)\to s)$$
$x+2=4-x\implies x=1.$
$a := x^2-y^3 \tag{*}\label{*}$
$a+y^3 \stackrel{\eqref{*}}= x^2$
$$\eqref{*}$$
$$\ref{*}$$
$$\eqref{*}$$
$(1)\qquad\qquad\sum\limits_{j}k\tag*{}$
$$(1)$$
$f\left(\left[ \frac{1+\left\{x,y\right\}}{\left(\frac{x}{y}+\frac{y}{x}\right)\left(u+1\right)}+a\right]^{3/2}\right).$
\begin{aligned} a=&\left(1+2+3+ \cdots \right. \\ & \cdots+ \left. \infty-2+\infty-1+\infty\right). \end{aligned}
$\left\langle q\middle\|\frac{\frac{x}{y}}{\frac{u}{v}} \middle| p \right\rangle.$
$$\Big(\dots\Big)$$
$\operatorname{arsinh}(x)$
$$arsinh(x)$$
$\operatorname*{Res}_{z=1}\left(\frac1{z^2-z}\right)=1$
$$\verb*{\rm ...}*$$
$$\verb*{\rm arsinh}*$$
$${\rm arsinh}$$
$$\operatorname{arsinh}x$$
$${\rm arsinh}x$$
$${\tt operatorname}$$
$$\text{arsinh }x$$
$$\operatorname{arsinh}x$$
$$\lim \limits_{x \to 1} \frac{x^2-1}{x-1}$$
$$\lim$$
$$lim$$
$$\lim \limits_{x \to 1}$$
$$\to$$
$$x \to 1$$
$$\lim$$
$$x \to 1$$
$$\lim \limits_{x \to 1} \frac{x^2-1}{x-1}$$
$\lim_{x\to 1}?$
$\mathop{+}\limits_{i=1}^k\text{ instead of }+_{i=1}^k$
$$\lim\limits_{x\to 1}$$
$$\lim_{x\to 1}$$
$$\lim_{x\to 1}$$
$$\lim\limits_{x\to 1}$$
$$x\to 1$$
$$\lim \limits_{x \to 1}$$
$$\lim\limits_{x\mapsto 1}\dfrac1x$$
$$\TeX$$
$$\lim_{x\mapsto 1}\frac1x$$
$$\TeX$$
$$\lvert x\rvert$$
$$\lVert v\rVert$$
$|x|, ||v|| \quad\longrightarrow\quad \lvert x\rvert, \lVert v\rVert$
$$\|x\|$$
$$\lVert x \rVert$$
$\bbox[yellow] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (1) }$
$\bbox[yellow,5px] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (1) }$
$\bbox[5px,border:2px solid red] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (2) }$
$\bbox[yellow,5px,border:2px solid red] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (1) }$
\begin{align} v + w & = 0 &&\text{Given} \tag 1\\ -w & = -w + 0 && \text{additive identity} \tag 2\\ -w + 0 & = -w + (v + w) && \text{equations $(1)$ and $(2)$} \end{align}
$\spadesuit\quad\heartsuit\quad\diamondsuit\quad\clubsuit$
$\color{red}{\heartsuit}\quad\color{red}{\diamondsuit}$
$♠\quad♡\quad♢\quad♣\\ ♤\quad♥\quad♦\quad♧$
$$\Rightarrow$$
$$\Leftarrow$$
$$\Leftrightarrow$$
$$\implies$$
$$\impliedby$$
$$\iff$$
$\require{enclose} \begin{array}{r} 13 \\[-3pt] 4 \enclose{longdiv}{52} \\[-3pt] \underline{4}\phantom{2} \\[-3pt] 12 \\[-3pt] \underline{12} \end{array}$
$x^3−6x^2+11x−6=(x−{\color{red}1})(x^2−5x+6)+{\color{blue}0}$
$\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 1 & -6 & 11 & -6\\ {\color{red}1} & \downarrow & 1 & -5 & 6\\ \hline & 1 & -5 & 6 & |\phantom{-} {\color{blue}0} \end{array}$
$$90°$$
$$90^\circ$$
$\sum_{n=1}^\infty \frac{1}{n^2} \to \textstyle \sum_{n=1}^\infty \frac{1}{n^2} \to \displaystyle \sum_{n=1}^\infty \frac{1}{n^2}$
$$\displaystyle \lim_{t \to 0} \int_t^1 f(t)\, dt$$
$$\lim_{t \to 0} \int_t^1 f(t)\, dt$$
$$\overline a+\overline b=\overline {a\cdot b}$$
$$\sqrt{a}-\sqrt{b}$$
$$\sqrt{\mathstrut a} - \sqrt{\mathstrut b}$$
$$\sqrt{\vphantom{b} a} - \sqrt{b}$$
$$\dfrac{1}{\sqrt{\vphantom{b} a} - \sqrt{b}}$$
$$\smash{\dfrac{1}{\sqrt{\vphantom{b} a} - \sqrt{b}}}$$
$$n$$
$$T_1,\ldots,T_n$$
$$T_1\gets 1$$
$$k$$
$$2$$
$$n$$
$$T_k\gets (k−1)T_{k−1}$$
$$k$$
$$2$$
$$n$$
$$j$$
$$k$$
$$n$$
$$T_j\gets (j −k)T_{j−1} + (j −k+2)T_j$$
$$\;T_1,T_2,\ldots,T_n$$
$$n$$
$$T_1,\ldots,T_n$$
$$T_1\gets 1$$
$$\;k\;$$
$$2\;$$
$$\;n$$
$$\phantom{{}++{}}$$
$$T_k\gets (k−1)T_{k−1}$$
$$\;k\;$$
$$2\;$$
$$\;n$$
$$\phantom{{}++{}}$$
$$\;j\;$$
$$\;k\;$$
$$\;n$$
$$\phantom{{}++{}}$$
$$\phantom{{}++{}}$$
$$T_j\gets (j −k)T_{j−1} + (j −k+2)T_j$$
$$\;T_1,T_2,\ldots,T_n$$
$$\LaTeX$$
$e=mc^2 \tag{1}\label{eq1}$
$$\eqref{eq1}$$
$$\eqref{eq1}$$
\begin{aligned} a &= b + c \\ &= d + e + f + g \\ &= h + i \end{aligned}\tag{2}\label{eq2}
$$\eqref{eq2}$$
$$\eqref{eq2}$$
\begin{align} a &= b + c \tag{3}\label{eq3} \\ x &= yz \tag{4}\label{eq4}\\ l &= m - n \tag{5}\label{eq5} \end{align}
$$\eqref{eq3}$$
$$\eqref{eq4}$$
$$\eqref{eq5}$$
\begin{array}{ll} \text{maximize} & c^T x \\ \text{subject to}& d^T x = \alpha \\ &0 \le x \le 1. \end{array}
\begin{alignat}{5} \max \quad & z = & x_1 & + & 12 x_2 & & & && \\ \mbox{s.t.} \quad & & 13 x_1 & + & x_2 & + & 12x_3 & \geq 5 && \tag{constraint 1} \\ & & x_1 & & & + & x_3 & \leq 16 && \tag{constraint 2} \\ & & 15 x_1 & + & 201 x_2 & & & = 14 && \tag{constraint 3} \\ & & \rlap{x_i \ge 0, i = 1, 2, 3} \end{alignat}
$$\max$$
$$z$$
$$\pm$$
$$\rm\LaTeX$$
$$\rm\LaTeX$$
$\begin{bmatrix} 1 & 2 & 2 \\ 2 & 3 & 4 \\ 4 & 4 & 2 \end{bmatrix}.$
$$z$$
$$z$$
\begin{array}{rrrrrr|r} & x_1 & x_2 & s_1 & s_2 & s_3 & \\ \hline s_1 & 0 & 1 & 1 & 0 & 0 & 8 \\ s_2 & 1 & -1 & 0 & 1 & 0 & 4 \\ s_3 & 1 & 1 & 0 & 0 & 1 & 12 \\ \hline & -1 & -1 & 0 & 0 & 0 & 0 \end{array}
\begin{array}{rrrrrrr|rr} & x_1 & x_2 & s_1 & s_2 & s_3 & w & & \text{ratio} \\ \hline s_1 & 0 & 1 & 1 & 0 & 0 & 0 & 8 & - \\ w & 1^* & -1 & 0 & -1 & 0 & 1 & 4 & 4 \\ s_3 & 1 & 1 & 0 & 0 & 1 & 0 & 12 & 12 \\ \hdashline & 1 & -1 & 0 & -1 & 0 & 0 & 4 & \\ \hline s_1 & 0 & 1 & 1 & 0 & 0 & 0 & 8 & \\ x_1 & 1 & -1 & 0 & -1 & 0 & 1 & 4 & \\ s_3 & 0 & 2 & 0 & 2 & 1 & -1 & 8 & \\ \hdashline & 0 & 0 & 0 & 0 & 0 & -1 & 0 & \end{array}
\begin{array}{rrrrrrrr|r} & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & \\ \hline x_4 & 0 & -3 & 7 & 1 & 0 & 0 & 2 & 2M -4 \\ x_5 & 0 & -9 & 0 & 0 & 1 & 0 & -1 & -M -3 \\ x_6 & 0 & 6 & -1 & 0 & 0 & 1 & -4^* & -4M +8 \\ x_1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & M \\ \hline & 0 & 1 & 1 & 0 & 0 & 0 & 2 & 2M \\ \text{ratio} & & & 1 & & & & 1/2 & \end{array}
\begin{array}{rrrrrrr|r} & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & \\ \hline s_1 & -2 & 0 & -2 & 1 & 0 & 0 & -60 \\ s_2 & -2 & -4^* & -5 & 0 & 1 & 0 & -70 \\ s_3 & 0 & -3 & -1 & 0 & 0 & 1 & -27 \\ \hdashline & 8 & 10 & 25 & 0 & 0 & 0 & 0 \\ \text{ratio} & -4 & -5/2 & -5 & & & & \\ \hline s_1 & -2^* & 0 & -2 & 1 & 0 & 0 & -60 \\ x_2 & 1/2 & 1 & 5/4 & 0 & -1/4 & 0 & 35/2 \\ s_3 & 3/2 & 0 & 11/4 & 0 & -3/4 & 1 & 51/2 \\ \hdashline & 3 & 0 & 25/2 & 0 & 5/2 & 0 & -175 \\ \text{ratio} & -3/2 & & 25/4 & & & & \\ \hline x_1 & 1 & 0 & 1 & -1/2 & 0 & 0 & 30 \\ x_2 & 0 & 1 & 3/4 & 1/4 & -1/4 & 0 & 5/2 \\ s_3 & 0 & 0 & 5/4 & 3/4 & -3/4^* & 1 & -39/2 \\ \hdashline & 0 & 0 & 19/2 & 3/2 & 5/2 & 0 & -265 \\ \text{ratio} & & & & & \dots & & \\ \hline x_1 & 1 & 0 & 1 & -1/2 & 0 & 0 & 30 \\ x_2 & 0 & 1 & 1/3 & 0 & 0 & -1/3 & 9 \\ s_2 & 0 & 0 & -5/3 & -1 & 1 & -4/3 & 26 \\ \hdashline & 0 & 0 & 41/3 & 4 & 0 & 10/3 & -330 \end{array}
$\require{extpfeil} % produce extensible horizontal arrows \begin{array}{ccc} % arrange LPPs % first row % first LPP \begin{array}{ll} \max & z = c^T x \\ \text{s.t.} & A x \le b \\ & x \ge 0 \end{array} & \xtofrom{\text{duality}} & % second LPP \begin{array}{ll} \min & v = b^T y \\ \text{s.t.} & A^T y \ge c \\ & y \ge 0 \end{array} \\ ({\cal PC}) & & ({\cal DC}) \\ \text{add } {\Large \downharpoonleft} \text{slack var} & & \text{minus } {\Large \downharpoonright} \text{surplus var}\\ % Change to your favorite arrow style % % second row % third LPP \begin{array}{ll} \max & z = c^T x \\ \text{s.t.} & A x + s = b \\ & x,s \ge 0 \end{array} & \xtofrom[\text{some steps skipped}]{\text{duality}} & % fourth LPP \begin{array}{ll} \min & v = b^T y \\ \text{s.t.} & A^T y - t = c \\ & y,t \ge 0 \end{array} \\ ({\cal PS}) & & ({\cal DS}) % \end{array}$
$$\sum_{i=n}^\mathbb{N}$$
$$\sum$$
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 Math.SE extracted Samples

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f(n) =
f(n) =