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 apache delimiters are double-dollar signs $$ ... $$ or escaped brackets \[ ... \] for displayed apache, and escaped parentheses \( ... \) or single-dollar signs $ ... $ for in-line apache. 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 apache 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*}