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Repeating fractions

1(ϕ5ϕ)e25π1+e2π1+e4π1+e6π1+e8π1+\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

(k=1nakbk)2(k=1nak2)(k=1nbk2)\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

i=1k+1i=(i=1ki)+(k+1)=k(k+1)2+k+1=(i=1ki)+(k+1)=k(k+1)2+k+1=k(k+1)+2(k+1)2=(k+1)(k+2)2=(k+1)((k+1)+1)2\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+q2(1q)+q6(1q)(1q2)+=j=01(1q5j+2)(1q5j+3), for q<1.1 + \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

udvdxdx=uvdudxvdx\int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx

Cross Product

V1×V2=ijkXuYu0XvYv0\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

(a11a12a13a21a22a23a31a32a33)\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{pmatrix} [0000]\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{bmatrix}

Case definitions

f(n)={n2,if n is even3n+1,if n is oddf(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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