Hello MathJax!!

The following equations are represented in the HTML source code as LaTeX expressions.
The Cauchy-Schwarz Inequality
\[\left(
\sum_{k=1}^n a_k b_k
\right)^2
\leq
\left(
\sum_{k=1}^n a_k^2
\right)
\left(
\sum_{k=1}^n b_k^2
\right)\]
Auto Numbering and Ref?
Suppose that \(f\) is a function with period $\pi$, then we have
$
\newcommand{\rd}{\rm d}
$
\begin{gather}\label{eq:1}
\int_t^{t+\pi} f(x)\rd x=\int_0^\pi f(x)\rd x.\tag{1}
\end{gather}
From \eqref{eq:1} we know that $\dots$.
Can we use \newcommand to define some macro?
$
\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2} \left( \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5} \right)}
$
define newcommand as:

$\newcommand{\Re}{\mathrm{Re}\,}
\newcommand{\pFq}[5]{{}_{#1}\mathrm{F}_{#2}
  \left(
    \genfrac{}{}{0pt}{}{#3}{#4} \bigg| {#5}
  \right)}$

We consider, for various values of $s$, the $n$-dimensional integral
\begin{align}\label{def:Wns}
W_n (s):=
\int_{[0, 1]^n}
\left| \sum_{k = 1}^n \mathrm{e}^{2 \pi \mathrm{i} \, x_k} \right|^s \mathrm{d}\boldsymbol{x}
\end{align}
which occurs in the theory of uniform random walk integrals in the plane, where at each step a unit-step is taken in a random direction. As such, the integral \eqref{def:Wns} expresses the $s$-th moment of the distance to the origin after $n$ steps.

By experimentation and some sketchy arguments we quickly conjectured and strongly believed that, for $k$ a nonnegative integer
\begin{align}\label{eq:W3k}
W_3(k)= \Re \, \pFq32{\frac12, -\frac k2, -\frac k2}{1, 1}{4}.
\end{align}
Appropriately defined, \eqref{eq:W3k} also holds for negative odd integers. The reason for \eqref{eq:W3k} was long a mystery, but it will be explained at the end of the paper.
How about MathML?
Definition of Christoffel Symbols

$${\left({\nabla }_{X}Y\right)}^k=X^i{({\nabla }_{i}Y)}^k=X^i(\frac{\partial Y^k}{\partial x^i}+{\Gamma }_{im}^k{Y}^m)$$

$${\left({\nabla }_{X}Y\right)}^k=X^i{({\nabla }_{i}Y)}^k=X^i(\frac{\partial Y^k}{\partial x^i}+{\Gamma }_{im}^k{Y}^m)$$