KaTeX
Repeating fractions
$$
\frac{1}{\Bigl(\sqrt{\phi \sqrt{5}}-\phi\Bigr) e^{\frac25 \pi}} \equiv 1+\frac{e^{-2\pi}} {1+\frac{e^{-4\pi}} {1+\frac{e^{-6\pi}} {1+\frac{e^{-8\pi}} {1+\cdots} } } }
$$
Summation notation
$$
\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)
$$
Sum of a Series
$$
\begin{aligned}
\sum_{i=1}^{k+1}i &= \left(\sum_{i=1}^{k}i\right) +(k+1) \\
&= \frac{k(k+1)}{2}+k+1 \\
&= \left(\sum_{i=1}^{k}i\right) +(k+1) \\
&= \frac{k(k+1)}{2}+k+1 \\
&= \frac{k(k+1)+2(k+1)}{2} \\
&= \frac{(k+1)(k+2)}{2} \\
&= \frac{(k+1)((k+1)+1)}{2}
\end{aligned}
$$
Product notation
$$
1 + \frac{q^2}{(1-q)}+\frac{q^6}{(1-q)(1-q^2)}+\cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})},\text{ for }\lvert q\rvert < 1.
$$
Calculus
$$
\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx
$$
Cross Product
$$
\mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\[2pt]
\frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\[4pt]
\frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0
\end{vmatrix}
$$
Matrices
$$
\begin{pmatrix}
a_{11} & a_{12} & a_{13}\\
a_{21} & a_{22} & a_{23}\\
a_{31} & a_{32} & a_{33}
\end{pmatrix}
$$
$$
\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}
$$
Case definitions
$$
f(n) = \begin{cases} \frac{n}{2}, & \text{if } n\text{ is even} \\ 3n+1, & \text{if } n\text{ is odd} \end{cases}
$$
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