Math & Fractions Showcase

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\newtheorem{definition}{Definition}

\title{Math \& Fractions Showcase}
\author{Bibby AI}
\date{\today}

\begin{document}
\maketitle

\section{Fractions}

Simple fraction: $\frac{a}{b}$

Display fraction: $$\dfrac{x^2 + 2x + 1}{x - 1}$$

Nested fractions:
$$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{ad}{bc}$$

Continued fraction:
$$e = 2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{4 + \cdots}}}}}$$

\section{Integrals}

Definite integral:
$$\int_0^1 x^2\,dx = \frac{1}{3}$$

The Gaussian integral:
$$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$$

Double integral:
$$\iint_D f(x,y)\,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx$$

\section{Summations and Products}

Geometric series:
$$\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \quad |r| < 1$$

Basel problem:
$$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

Factorial as product:
$$n! = \prod_{k=1}^{n} k$$

\section{Limits}

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$$

\section{Matrices}

$$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}, \quad
I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

Determinant:
$$\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$

\section{Aligned Equations}

\begin{align}
    (x + y)^2 &= x^2 + 2xy + y^2 \\
    (x - y)^2 &= x^2 - 2xy + y^2 \\
    (x + y)(x - y) &= x^2 - y^2
\end{align}

\section{Theorems}

\begin{theorem}[Pythagorean Theorem]
For a right triangle with legs $a$, $b$ and hypotenuse $c$:
$$a^2 + b^2 = c^2$$
\end{theorem}

\begin{definition}
A function $f: \mathbb{R} \to \mathbb{R}$ is \emph{continuous} at $x_0$ if
$$\lim_{x \to x_0} f(x) = f(x_0).$$
\end{definition}

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