# Inserting $\LaTeX$ in Your Page

You may insert $\LaTeX$ and it will be auto-typeset, here is an example. If you inspect the source code of the example below, you'll see that the math delimiters are double-dollar signs $$...$$ or escaped brackets $...$ for displayed math, and escaped parentheses $...$ or single-dollar signs $...$ for in-line math. For example inserting \$X\$ or \$X_q\$ into your HTML typesets as $X$ or $X_q$. When using a dollar sign as a currency symbol, be sure to escape it \\$. Do not insert global commands or preamble commands, only insert self-contained $\LaTeX$ snippets.

$\LaTeX$ is rendered through the MathJax engine. If you create your web page following our set-up, this feature is automatically incorporated. For more information, right-click on any of the displayed math below.

Proof of the identity

$$\int_0^1\frac{dx}{x^x}=\sum_{n=1}^\infty \frac1{n^n}=\frac1{1^1}+\frac1{2^2}+\frac1{3^3}+\dots\ .$$ First, recall the integral $$\Gamma(n)\equiv\int_0^\infty e^{-s}s^{n-1}\,ds=(n-1)!$$ Then use the substitution $x=e^{-t}$, the exponential series, shifting the index $n$ by one, the substitution $nt=s$, and $\Gamma(n)=(n-1)!$: \begin{eqnarray*} \int_0^1x^{-x}\,dx &=& \int_0^\infty \left(e^{-t}\right)^{-e^{-t}}e^{-t}\,dt =\int_0^\infty e^{te^{-t}}e^{-t}\,dt\\ &=& \int_0^\infty \left[\sum_{n=0}^\infty \frac1{n!}t^ne^{-nt}\right]e^{-t}\,dt\\ &=& \sum_{n=0}^\infty\frac1{n!}\int_0^\infty t^ne^{-(n+1)t}dt\\ &=& \sum_{n=1}^\infty\frac1{(n-1)!}\int_0^\infty e^{-nt}t^{n-1}\,dt\\ &=& \sum_{n=1}^\infty\frac1{(n-1)!}n^{-n}\int_0^\infty e^{-s}s^{n-1}\,ds\\ &=& \sum_{n=1}^\infty\frac1{(n-1)!}n^{-n}\Gamma(n)= \sum_{n=1}^\infty n^{-n}. \end{eqnarray*}