Testing latex

$$\int_{-\infty}^{\infty} e^{-x^2/2} , dx = \sqrt{2\pi}$$$$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$$$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$$$$\begin{vmatrix}a & b \\c & d\end{vmatrix} = ad - bc$$$$\frac{d}{dx}\left(\frac{x^2 + 1}{x - 1}\right)$$$$\iiint_V (\nabla \cdot \mathbf{F}) , dV = \oint_{\partial V} \mathbf{F} \cdot d\mathbf{S}$$$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$$$\ln\left(\frac{f(x)}{g(x)}\right) = \ln f(x) - \ln g(x)$$$$\forall x \in \mathbb{R}, \exists y \in \mathbb{R} \text{ such that } x + y = 0$$$$\sqrt{\frac{x^2 + y^2}{x^2 - y^2}}$$$$\begin{array}{c|c}A & B \\hlineC & D\end{array}$$$$\sum_{i=1}^n \sum_{j=1}^n a_{ij}x_i x_j$$$$\mathcal{L}{f(t)}(s) = \int_0^\infty e^{-st}f(t),dt$$$$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$$$\mathbf{A} = \begin{pmatrix}a_{11} & a_{12} \\a_{21} & a_{22}\end{pmatrix}, \quad\mathbf{B} = \begin{pmatrix}b_{11} & b_{12} \\b_{21} & b_{22}\end{pmatrix}$$$$\underbrace{a + b + \dots + z}{26}$$$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$$$\langle \psi | \phi \rangle = \int{-\infty}^{\infty} \psi^*(x)\phi(x) , dx$$$$\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$$$$\prod_{k=1}^n \left(1 + \frac{1}{k}\right) = \frac{(n+1)}{1}$$$$S(\omega)=1.466, H_s^2 \frac{\omega_0^5}{\omega^6} \exp\Bigl[-3^{\frac{\omega}{\omega_0}}\Bigr]^2$$