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#include<bits/stdc++.h>

$$\sum_{i=1}^n i = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2} $$$$\sum_{i=1}^n i^2 = 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6} $$$$\sum_{k=0}^n (a + kd) = a + (a+d) + (a+2d) + \cdots + (a+nd) = \frac{(n+1)(2a + nd)}{2} $$$$\sum_{k=0}^\infty ar^k = a + ar + ar^2 + \cdots = \frac{a}{1 - r} $$$$\sum_{k=0}^\infty \frac{x^k}{k!} = e^x$$ $$\sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!} = \sin x $$$$\sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} = \cos x $$$$\int_a^b f(x) , dx = \lim_{n \to \infty} \sum_{i=1}^n f\left( a + i \cdot \frac{b-a}{n} \right) \cdot \frac{b-a}{n} $$$$E[X] = \sum_{x} x \cdot P(X = x)$$ $$\text{Var}(X) = \sum_{x} (x - E[X])^2 \cdot P(X = x) $$$$\sum_{i=1}^n \sum_{j=1}^m a_{ij} = \sum_{j=1}^m \sum_{i=1}^n a_{ij} $$$$\sum_{i=1}^n \sum_{j=1}^i f(i,j) = \sum_{j=1}^n \sum_{i=j}^n f(i,j) $$$$\sum_{i=1}^n i = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2} $$$$\sum_{i=1}^n i^2 = 1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6} $$$$\sum_{k=0}^n (a + kd) = a + (a+d) + (a+2d) + \cdots + (a+nd) = \frac{(n+1)(2a + nd)}{2} $$$$\sum_{k=0}^\infty ar^k = a + ar + ar^2 + \cdots = \frac{a}{1 - r} $$$$\sum_{k=0}^\infty \frac{x^k}{k!} = e^x$$ $$\sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k+1)!} = \sin x $$$$\sum_{k=0}^\infty (-1)^k \frac{x^{2k}}{(2k)!} = \cos x $$$$\int_a^b f(x) , dx = \lim_{n \to \infty} \sum_{i=1}^n f\left( a + i \cdot \frac{b-a}{n} \right) \cdot \frac{b-a}{n} $$$$E[X] = \sum_{x} x \cdot P(X = x)$$ $$\text{Var}(X) = \sum_{x} (x - E[X])^2 \cdot P(X = x) $$$$\sum_{i=1}^n \sum_{j=1}^m a_{ij} = \sum_{j=1}^m \sum_{i=1}^n a_{ij} $$$$\sum_{i=1}^n \sum_{j=1}^i f(i,j) = \sum_{j=1}^n \sum_{i=j}^n f(i,j) $$$$\int^{\infty}_{-\infty}e^{{-x}^2}dx=\sqrt\pi$$ $$\sum_{k=0}^n ar^k = a \frac{1-r^{n+1}}{1-r} \quad (r \neq 1) $$$$\sum_{k=1}^n \frac{1}{k} \approx \ln n + \gamma \quad $$$$\sum_{i=1}^n i^3 = \left( \frac{n(n+1)}{2} \right)^2 $$$$\int_a^b f(x) , dx = F(b) - F(a) \quad$$ $$\int u , dv = uv - \int v , du$$ $$f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} (x-a)^k $$$$\int_{-\infty}^\infty e{-ax2 + bx} , dx = \sqrt{\frac{\pi}{a}} e{\frac{b2}{4a}} \quad (a>0) $$$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$ $$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e{-\frac{(x-\mu)2}{2\sigma^2}} $$$$\text{Cov}(X,Y) = E[(X-E[X])(Y-E[Y])]$$ $$\det \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc $$$$\det(A - \lambda I) = 0 \quad$$ $$\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i $$$$e^{i\theta} = \cos \theta + i \sin \theta$$ $$a^{p-1} \equiv 1 \pmod{p} \quad$$ $$\pi(n) \sim \frac{n}{\ln n} \quad$$ $$y’ + P(x)y = Q(x) \implies y = e^{-\int P , dx} \left( \int Q e^{\int P , dx} , dx + C \right) $$$$\frac{\partial u}{\partial t} = \alpha \nabla^2 u$$ $$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} , dt \quad $$$$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} \quad$$ $$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$ $$F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$$ $$E=mc^2$$ $$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V \psi $$$$i\hbar\gamma^\mu\partial_\mu\psi=mc\psi$$ $$\Alpha \alpha \Beta \beta \Gamma \gamma \Delta \delta \Epsilon \epsilon \Zeta \zeta \Eta \eta \Theta \theta \Lambda \lambda \Xi \xi \Pi \pi \Sigma \sigma \Phi \phi \Psi \psi $$$$\Omega \omega \R\Z$$