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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:

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

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