$\frac{(n-1)^2}2<\frac{n(n-1)}2<\frac{n^2}2<\frac{n(n+1)}2<\frac{(n+1)^2}2$


$E|X| \le \mathbb{E} |X|^3  $


$e_i \mapsto \lambda_i e_i, 1 \le i \le n.$


$\int_{-\pi}^{\pi}v(x)\cos nx\mathbb dx=\int_{-\pi}^{\pi}v(x)\sin nx\mathbb dx=0$


$U,V\subseteq A$


$s\cdot s \equiv 1 \Rightarrow s'\cdot s + s\cdot s'\equiv 0 \Rightarrow s\cdot s'\equiv 0.$


$\frac{a+b x}{x^5+x+1}$


$x+\left(\frac{dy}{dx}\right)= \sqrt{1+\left(\frac{dy}{dx}\right)^2 }$


$P(i) \implies P(i+1).$


$x_n ={n(n+1) \above 1.5pt 2}$


$\Delta\subseteq V\subseteq U$


$\sum_{n=1}^\infty e^{-\pi n^2x}\le e^{-\pi x}\sum_{n=0}^\infty e^{-\pi nx} $


$\vdash (\forall x,\forall y :  (xRy \lor yRx)) \Rightarrow (\forall x : xRx)$


$\forall n>0 : C (n) \implies C (n+1)$


$\int _{0}^{1}{\tan^{-1}\left(x\right) \over {1+x}}dx$


$ \bbox[lightyellow] {   P_{\,M}(r,p,n) = \sum\limits_{0\, \le \,\,s\,\, \le \,n} {p^{\,s} \left( {1 - p} \right)^{\,n - s} \sum\limits_{\left( {0\, \le } \right)\,\,k\,\,\left( { \le \,{s \over r}\, \le \,n - s + 1} \right)} {\left( { - 1} \right)^k \left( \matrix{   n - s + 1 \cr    k \cr}  \right)\left( \matrix{   n - k\left( {r + 1} \right) \cr    s - k\left( {r + 1} \right) \cr}  \right)} }  }$


$\sum_{n\ge 1}\frac1n=\sum_{n\ge 1}\frac1{2n}+\sum_{n\ge 1}\frac1{2n+1}\;.$


$J(\gamma) = (\gamma - 1, \gamma)$


$1 + 2 + 3 + 4 + 5 + \cdots = -{\frac{1}{12}}$


$\left\{ \begin{array}{c}n+1\\k-1\end{array} \right\}\left\{ \begin{array}{c}n+1\\k+1\end{array} \right\}<\left\{ \begin{array}{c}n+1\\k\end{array} \right\}^2$


$C=\{\langle x,y\rangle\in X\times y:xy\ge 1\}\;;$


$log_{b^{n}} x = \frac {1}{n} log_{b} x$


$S \subseteq \cup_{\gamma \in \Gamma}\ U_\gamma$


$E_2^{2,0}=E_2^{0,2}\rightarrow E_3^{0,2} \rightarrow \ldots\rightarrow E_{\infty}^{0,2}$


$aX_0 + bY_0 = d$


$n!=\sqrt{2\pi n}(n/e)^n\left(1+O\left(\frac{1}{n}\right)\right)$


$           \int_a^b f(t)dt  = \int_a^b t f(t)dt = 0 $


$(\frac{\gamma u}{(1+\gamma)c})^2 = \frac{\gamma^2-1}{(\gamma+1)^2} = \frac{\gamma-1}{\gamma+1}$


$ \left\{ \matrix{   P_{\,0} (x) = x,\quad P_{\,1} (x) = x^{\,2}  \hfill \cr    P_{\,n + 2} (x) = x^{\,2} P\,'_{\,n + 1} (x) + \left( {n + 1} \right)x\,P_{\,n + 1} (x) - \left( {n + 1} \right)P\,'_{\,n} (x) \hfill \cr}  \right. $


$\mathrm{d}S=\sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2}\mathrm{d}x$


$\displaystyle f(1-k)=\frac{9^{1-k}}{9^{1-k}+3}=\frac9{9+3\cdot9^k}=\frac3{9^k+3}$


$|f(x)| \le \max_{z \in A} |f(z)|$


$\begin{pmatrix}a & -\overline{b} \\ b & \overline{a} \end{pmatrix}.$


$f(a+b)\le f(a)+f(b)$


$P[X_{n+1} \in S' | X_n = s] = 1$


$\quote(+A+\quote\to+B+\quote)$


$f(f(x)^2 y) = x^3 f(xy)$


$\frac{I}{kq} = \int_{0}^{1}\frac{1}{(x^2+1)\sqrt{x^2+2}}\,dx$


$\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}$


$A_0 \subseteq A_1 \subseteq A_2 \subseteq \dots$


$\lim_{x\to x_0^-}f'(x)=\lim_{x\to x_0^+}f'(x)=l$


$\mathbb{R}^n=\mathbb{R}\times\mathbb{R}^{n-1}$


$\lim_{n \rightarrow \infty} \mu (A_n) = \mu (\bigcup A_n)$


$(a+b, a-b) \ge (a, b)$


$\lim_ {n\to \infty}\ \frac{1}{2^n}$


$\sum_{\gamma<\alpha+1} a_\gamma=\sum_{\gamma<\alpha} a_\gamma + a_\alpha \le \prod_{\gamma<\alpha} a_\gamma + a_\alpha \overset{(1)}\le \prod_{\gamma<\alpha+1} a_{\gamma}.$


$P(X_t=3|X_0=1)=1-e^{-\mu t}$


$\displaystyle \lim_{x \to c^-} f(x) < \lim_{x \to c^+} f(x)$


$ \begin{align} \int_0^{\pi/2}\frac{x\log(\sin(x))}{\sin(x)}\,\mathrm{d}x &=\int_0^1\frac{\sin^{-1}(u)}{\sqrt{1-u^2}}\frac{\log(u)}u\,\mathrm{d}u\\ &=\int_0^1\sum_{k=0}^\infty\frac1{(2k+1)}\frac{4^k}{\binom{2k}{k}}u^{2k}\log(u)\,\mathrm{d}u\\ &=-\sum_{k=0}^\infty\frac1{(2k+1)^3}\frac{4^k}{\binom{2k}{k}}\tag{3} \end{align} $


$z=( \alpha_{1},..., \alpha_{n},  \beta_{1},....,\beta_{n})$


$(f+g)(x+ab)=f(x+ab)+g(x+ab)=f(x)+g(x)=(f+g)(x)$


$\dfrac{p}{1-(\varepsilon\cos\,\varphi)^2}$


$  \alpha_2^{-} = \frac{ (1-f)^3 }{ 1 + ( 1 - f)^2 } $


$ax+by=t$


$\begin{pmatrix}     4    & -1\\     2    & 1  \\   \end{pmatrix}$


$\sum_{n=m+1}^{\infty}\left\|e_{n}-f_{n}\right\|^{2}<1. \tag{III}$


$\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$


$\det\begin{pmatrix} A & B\\ B & A \end{pmatrix}=\det(A-B)\det(A+B).$


$F_\nu (\xi) = \frac{1}{\Gamma(\nu)}\int_0^\infty{\frac{x^{\nu-1}}{e^x/\xi -1}dx}$


$U\cap L=\{\langle x,y\rangle\in L:x_0\le x<b\}\;,$


$\|x^i_n-w_i\|\le \epsilon$


$p_1p_2...p_n +1$


$\alpha, \omega \alpha, \omega^{2} \alpha$


$\frac{|\langle x,y\rangle|}{||x||\cdot||y||}$


$\mathbf x = (x,y),$


$X_1 \subseteq X_2 \subseteq \cdots$


$f_{Z}(z)=f_{U}(u)=\frac{1}{2}(1-e^{\frac{-1}{a}})e^{\frac{-1}{a}(u)}$


$P[X_i \geq 0] = 1$


$ \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x).$


$\left\lbrace \begin{array}{ll}  y=\alpha(2x) \\  x=\alpha(2y) \\  x^2+y^2-2=0 \end{array} \right.$


$R = \{\begin{bmatrix}x \\y\end{bmatrix}\in \mathbb R^2:a \le x\le b \, \, , c\le y \le d\,  \}$


$|I_j| = (b_j - a_j) = |\bar{I_j}|$


$\hat{\mu} = \frac{x_{1} + x_{2} + ... + x_{n}}{n}$


$ \left[   \begin{array}{ccc|c}     3 & 1 & 1 & 14\\     -1 & -2 & -3 & -9\\     5 & -1 & 5 & 30   \end{array} \right] $


$\log_a(bc) = \log_a(b) \log_a(c)$


$ ds = \sqrt{1+\bigg(\frac{dy}{dx}\bigg)^2}\, dx. $


$ d(x,y)= \frac{||x-y||}{1+||x-y||}$


$P(3) \implies P(4),$


$ N_{\,1} (n + 1) = N_{\,1} (n) + N_{\,1} (n - 1)\quad \left| \matrix{   \;N_{\,1} (0) = 0 \hfill \cr    \;N_{\,1} (1) = 1\; \hfill \cr}  \right. $


$m \equiv 1 \mod p$


$z=\sqrt[4]{2}e^{\frac{7\pi i}{6}}$


$\Pi^{+}$


$\begin{align}\mathsf E(XY^j\mid XY^j<w) ~=~& \dfrac{\mathsf E (XY^j~\mathbf 1_{XY^j<w}) }{ \mathsf P(XY^j<w)} \\[1ex]~=~&\dfrac{\mathsf E (X~\mathsf E(Y^j~\mathbf 1_{Y<(w/X)^{1/j}}\mid X)) }{ \mathsf P(Y<(w/X)^{1/j})} &\text{only if}~&j > 0 \\[1ex] ~=~& \bbox[gainsboro]{\dfrac{\int\limits_0^\infty~x~\mathsf E( Y^j~\mathbf 1_{Y<(w/x)^{1/j}}\mid X=x)~f_X(x)~\mathrm d x }{\int\limits_0^\infty~\mathsf P(Y<(w/x)^{1/j}\mid X=x)~f_X(x)~\mathrm d x }} \\[1ex] ~=~& \dfrac{\int\limits_0^\infty~x~f_X(x)\int\limits_0^{(w/x)^{1/j}} y^j~f_{Y\mid X}(y\mid x)~\mathrm d~y~\mathrm d x }{\int\limits_0^\infty~f_X(x)\int\limits_0^{(w/x)^{1/j}} f_{Y\mid X}(y\mid x)~\mathrm d~y~\mathrm d x } \end{align}$


$ x'+y'=(x'+y)'=(y+x')'=(y+x)''=(x+y)''=(x+y')'=(y'+x)'=y'+x'; $


$f(A+B)=f(A)+f(B).$


$\frac{x_1+x_2+...+x_n}{n}\geq \sqrt[n]{x_1x_2...x_n}$


$f(r) = 1-r -\dfrac{1-r^m}{(1+r)^m}=(1-r)\left(1-\dfrac{1}{(1+r)^m}-\dfrac{r^2}{(1+r)^m}-\cdots-\dfrac{r^{m-1}}{(1+r)^m}\right)>0 \iff (1+r)^m > 1+r+r^2+\cdots +r^{m-1}$


$ = \left[     \begin{array}{ccc|c}       1&-2&3&2\\       0&3&-2&k-2\\       0&0&0&k^2-k-6     \end{array} \right] $


$ \pi_* ( F_* m ) = ( \pi \circ F)_* m = (f \circ \pi)_* m = f_* ( \pi_* m) =f_* \mu = \mu $


$ \sum x_n >0$


$f;g: A_1\to A_1$


$\displaystyle{\frac{b \gamma_2}{\gamma_1}=\lambda=\frac{a \gamma_1}{\gamma_2}}$


$\frac{1-r^N}{1-r}=1+r+r^2+r^3+\dots+r^{N-1}$


$B = \{ \{ a\}, a  \}$


$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^4(m^2+n^2)}$


$M_1=\left(\begin{array}{} A & B\\ B &A \end{array}\right)$


$\lVert T_x\rVert=\lVert x\rVert$


$y_n=\frac{x_1+x_2+⋯+x_n}{n}\to l$


$I_n:=\int_{0}^{\pi}(\sin x)^ndx$


$d=\{(x,y)\in\mathbb{R}^2\ |\ 0 \le x \le 3, 0 \le y \le 2\}$


$\left\vert a-b \right\vert = r.$


$a'x+b'y=d$


$\det (A) = \left[a+(n-1)b\right](a-b)^{n-1}$


$\int_{-\pi}^\pi\frac{\sin(nx)}{2^n}dx=0,$


$\forall\varepsilon\gt0,\quad \exists\delta\gt0,\quad \underline{\forall x\in D},\quad |x-x_0|\lt\delta\implies|f(x)-f(x_0)|\lt\varepsilon.$


$\psi(a)=\big\{\{a\}\big\}$


$\left[\begin{array}{rrr|r} 2 & 1 & 1 & 5\\ 1 & -1 & 1 & 3 \\ -2 & p & 2 & q\end{array}\right]$


$\overline{\mathbb{R}}\times\overline{\mathbb{R}}$


$ Y =   \begin{pmatrix}   A & -B \\   B & A  \\   \end{pmatrix}. $


$\mid f(z^2)\mid\geq \mid f(z)\mid$


$||x^{k+1} - y^{k+1}|| \leq \int_{t=0}^{1}||F'(x^{k})^{-1}\left( F'(x^{k}) - F'(y^{k} + t(x^{k} - y^{k})) \right) (x^{k} - y^{k}) \text{d}t|| + \\ \quad \quad \quad ||F'(x^{k})^{-1}(F'(y^{k}) - F'(x^{k}))\Delta y^{k}||$


$p'(m')=p'(m)$


$ \int_{-\pi}^\pi \frac {\sin(n\theta)} {\sin(\theta)}\,d\theta. $


$\Bbb R^n \wedge \Bbb R^n$


$A\subset B\subset \bar B\subset U$


$ \sin(\pi s) \Gamma(s) = \frac{i}{2} \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}  \oint_C \frac{(-x)^{s-1}}{e^x-1} dx . \qquad (*)  $


$ N(n)=n^2-n+2 $


$\mathbb ET = |a|b$


$c_r(E) = e(E_{\mathbb R})$


$(y)^{y'}=(y')^{y}$


$\frac{1}{x} + \frac{1}{y} = \frac{1}{a} + \frac{1}{b}$


$\chi_{A_I}:= \prod_{j=1}^n \chi_{A_j}^{I_j}.$


$N = \lfloor \frac{x-p}{\delta}\rfloor \leq \lfloor \frac{b-a}{\delta}\rfloor$


$=\dfrac1{2x}\int\dfrac{2x}{(x^2+1)^n}dx-\int\left(\dfrac{d(1/2x)}{dx}\int\dfrac{2x}{(x^2+1)^n}dx\right)dx=?$


$ \delta A (x+\delta x)+A\delta x=0, $


$f'(x) = \frac{2}{3}e^{\frac{x}{3}} + \frac{1}{2}e^{-\frac{x}{2}} > 0 $


$\|A\|_2 = \sqrt{\lambda _{\max}(A^TA)}$


$r (f(x)+g(x))=r f(x) + r g(x)$


$P[X_n\geq 1/2]=1$


$ \mu\left(\bigcup_{n\geq 1}A_n\right)=\lim_{n\to\infty}\mu(A_n). $


$ F(x)=\int_a^xf(t)dt-\frac{a+x}{2}\int_a^xf(t)dt. $


$p_1, p_2 \mid b$


$c =|[a,b]| \leq |\mathbb{R}  - \{r\}| \leq | \mathbb{R}| = c$


$\lim_{n \rightarrow \infty} \frac{a_{1}+a_{2}+...+a_{n}}{n} = A$


$\left(\frac{\partial^2}{\partial x^2} +2 \frac{\partial}{\partial x}\right) \{i\sin(x)+\cos(x)\} = (-1+2i)(i\sin(x)+\cos(x))$


$\tan\mu = \frac{\mu}{2}$


$1 + \frac{1}{2^n - 2} = \frac{2^n - 1}{2^n - 2} = \frac{1}{2} \frac{2^n - 1}{2^{n-1} - 1}$


$f_X(x) = \theta e^{- \theta x},\ x >0$


$A_1 \subset A_2 \subset \dots \subset A_k \subset A_{k+1} \subset ...$


$ |f(y_1)|\le\frac{1}{2}|f(x_1)|$


$ J_n= \int_0^\frac{\pi}{2} \frac{\sin(2nx)}{\sin x}\:\mathrm{d}x,\quad I_n = \int_0^\frac{\pi}{2} \frac{\sin(2n+1)x}{\sin x}\:\mathrm{d}x $


$F = A \frac{(1+i)^{2n}-(1+i)^n}{i(1+i)^n}$


$\dfrac3a + \dfrac4b = 1$


$\frac{\pi^2}{6} = \sum_{n=1}^\infty \frac{1}{n^2} = \sum_{n=1}^\infty \frac{1}{(2n)^2} +\sum_{n=0}^\infty \frac{1}{(2n+1)^2} $


$I_n = \int_{-\infty}^\infty \frac{|\sin \left( \frac{x}{n} \right) \sin(x)|}{\frac{x}{n} x} \, dx.$


$n!+n \geq n!$


$ \langle a \wedge b, c \wedge d \rangle := \begin{vmatrix} \langle a,c \rangle & \langle a,d \rangle \\ \langle b,c \rangle & \langle b,d \rangle \end{vmatrix} $


$ 2\pi \int x ds = 2\pi \int x \sqrt{dx^2 + dy^2} $


$F(A+B)\cong F(A)+F(B)$


$ \begin{cases} x=-2s\\ y=0\\ z=s\\ \end{cases} $


$ \sqrt{1+  \left(\frac{dy }{ dx} \right)^2}= \frac{d^2 y }{dx^2} $


$\displaystyle S(n) = \sum_{k = 1}^n k = \dfrac{n(n+1)}{2}$


$f=\{\{\{a\},\{a,f\}\}\}$


$I_n=\int \frac{\sin(2nx)}{\sin x}dx$


$\int\frac{e^x \sin(2x)}{\cos^2(2x)}\mathrm{d}x = \int\frac{e^x 2\sin(x)\cos(x)}{(cos^2(x)-sin^2(x))^2}\mathrm{d}x=\int\frac{e^x 2\sin(x)\cos(x)}{(\cos(x)-\sin(x))^2(\cos(x)-\sin(x))^2}\mathrm{d}x=\int\frac{e^x 2\sin(x)\cos(x)}{(1 +2\cos(x)\sin(x))(1-2\cos(x)\sin(x))}\mathrm{d}x$


$A_1 \subseteq A_2 \subseteq \cdots \subseteq A_m \subseteq \cdots$


$\int \frac{\gamma^{2 x} dx}{\gamma^x + 1} = \frac{1}{\log \gamma}(\gamma^x - \log(\gamma^x + 1)) + C .$


$n\le f_j(x) < n+1$


$\lim_{n\to\infty}\frac{1+(\sqrt{n}+1)^{3}+2\sqrt{n}}{n+\sin(n)}$


$x*(y*z) = x*(y+z+2) = x + (y+z+2) + 2 = (x+y+2) + z + 2 = (x*y)*z$


$\|A\|_2=\sqrt{\max_n\lambda_n(AA^t)} $


$(\mathbb{R}^n\times\mathbb{R}^n,\mathcal{L}(\mathbb{R}^n)\times\mathcal{L}(\mathbb{R}^n))$


$a_n= \frac{n (n + 1) (2n + 1)} {6}$


$\lim_{n \rightarrow \infty}m(A_n) =m(A)$


$\begin{pmatrix}A & -B \\ B & A \end{pmatrix}$


$\int \frac{x}{({x^2+1})^6}dx$


$k=2 \rightarrow \left[\begin{array}{ccc}1&0&1\end{array}\right], k=3 \rightarrow \left[\begin{array}{ccc}1&1&1\end{array}\right]$


$Y_n = \frac{X_1 + X_2 + \ldots + X_n}{n} - n$


$f(x;\,\theta)=\frac{2x}{\theta^2}\qquad \text{for }0<x<\theta.$


$ax+by+cy=d$


$\int_0^1 \frac{\sin^4(x)}{x^2}$


$ (x,(y,z)) = (x,\{\{y\},\{y,z\}\})=\{\{x\},\{x,\{\{y\},\{y,z\}\}\}\}$


$\tag{2} x^5+1=(x+1)(x- e^{\frac{\pi i}{5}})(x- e^{\frac{-\pi i}{5}})(x- e^{\frac{3\pi i}{5}})(x- e^{\frac{-3\pi i}{5}}).$


$ x * y = y*(y*(x*y)) = y * ((x * (x*y))*(x*y)) = y *x. $


$f(\mathbf{x})=\sum^N_{k=1} a_k\chi_{E_k}(\mathbf{x})$


$d'(a) = b'(b'(b'(a))) = b'(b'(b)) = b'(c) = d\\ d'(b) = b'(b'(b'(b))) = b'(b'(c)) = b'(d) = a\\ d'(c) = b'(b'(b'(c))) = b'(b'(d)) = b'(a) = b\\ d'(d) = b'(b'(b'(d))) = b'(b'(a)) = b'(b) = c.$


$S=(\gamma e)^{\frac{1}{\gamma}}\Gamma_L(1+\frac{1}{\gamma},\frac{1}{\gamma})$


$\left(\begin{array}{rr} a& -b\\b&a\end{array}\right),$


$F_{1/3}(x)=\int\frac{dx}{\left(1+x^{1/3}\right)^{1/3}}=\frac9{40}\left(9-6x^{1/3}+5x^{2/3}\right)\left(1+x^{1/3}\right)^{2/3}$


$|X_n-X|^{2r} \leq 2^{2r} (|X_n|^{2r}+|X|^{2r}),$


$x*y*z+x'*y*z+x*y'*z+x*y*z' = y*z+x*z+x*y$


$A_1 \times \ldots \times A_k$


$\lim_{n\to \infty}\sum_{i=1}^n  \frac{1}{n+i}$


$ \int\frac{1}{(1+x^2)^k} dx\qquad(1)$


$P(X_2 \leq x \mid X_1 = x)\to0$


$dy/dx=f(x,y), R=\{(x, y):a\le x\le b, c \le y \le d\}$


$\forall \epsilon > 0, \exists \delta > 0 \text{ s.t. } |x - a| < \delta \implies |f(x) - f(a)| < \epsilon$


$\gamma_a \gamma_b = \gamma_{ab} = \gamma_{ba} = \gamma_b \gamma_a$


$ \frac{d}{dx}( \frac{1}{n}(1-e^{-x})^{n})=(1-e^{-x})^{n-1}e^{-x}$


$f(n+2)=2(f(n+2))^2 -f(n+2)f(n)-2012$


$ \begin{array} {|c|c|} \hline M^1                  & \begin{bmatrix} 3 & 1 \\ 5 & 3 \end{bmatrix} \\ \hline M^2 = (M^1)^2        & \begin{bmatrix} 14 & 6 \\ 30 & 14 \end{bmatrix} \\ \hline M^4 = (M^2)^2        & \begin{bmatrix} 76 & 68 \\ 40 & 76 \end{bmatrix} \\ \hline M^8 = (M^4)^2        & \begin{bmatrix} 96 & 36 \\ 80 & 96 \end{bmatrix} \\ \hline M^{16} = (M^8)^2       & \begin{bmatrix} 96 & 12 \\ 60 & 96 \end{bmatrix} \\ \hline M^{32} = (M^{16})^2    & \begin{bmatrix} 36 & 4 \\ 30 & 36 \end{bmatrix} \\ \hline M^{34} = M^{32}M^2     & \begin{bmatrix} 24 & 70 \\ 60 & 24 \end{bmatrix} \\ \hline \end{array}$


$f'(x) = \sqrt{(\frac {dL}{dx})^2 - 1}$


${\mathbb R}^{n+1}-\{0\}$


$ \mathbb{E} [X^2] \geq \mathbb{E} [|X|]^2 $


$M M M M M$


$y = 4*[x^9*x^6]$


$D^+$


$\int_0^\infty F(x) ~ dx$


$\overline H:=\bigl\{\langle x,y\rangle\in\Bbb R^2:x\ge0\bigr\},$


$A\subseteq B\subseteq\overline B\subseteq U\cap B(0,r)\subseteq U$


$f(t)=\frac1\pi \int_0^\pi \frac{\cos(xt)}{\sqrt{1+\sin^2(x/2)}}\,dx \tag 1$


$P[X\in A|\mathcal{F}_t] = P[X\in A|\mathcal{F}_s]$


$4^n/(4^n+2 )$


$\forall \mathbf{x}\in\mathbb{R}^{n+1}$


$(X,Y)\mapsto [X,Y]=XY-YX$


$\left \lfloor{\frac{1}{n}\sum_{k=1}^n\sqrt{k}}\right \rfloor=\left \lfloor{\left(\frac{2}{3}+\frac{1}{6n}\right)\sqrt{n+1}}\right \rfloor$


$\frac{\left( 1+\tan x\right) ( 1+\tan ^{2}x) }{(1-\tan x) \left( 1+\tan ^{2}x\right) }=\frac{(1+\tan x)^{2}\left( 1-\tan x\right) }{\left( 1-\tan x\right) \left( 1+\tan ^{2}x\right) },\qquad |\tan x|< 1,$


$ a\equiv b^s, $


$(3)(9)^{\frac{1}{3}}(27)^{\frac{1}{9}}...=x^3$


$f(x) = \frac{5}{x^2}$


$\mathbb R^n- \mathbb R$


$G(x) = \int_a^x F(t)dt$


$\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$


$r = a^{\log_a(r)}$


$\Phi^+$


$\left\lfloor \frac{ \left\lfloor x/2 \right\rfloor }{ 2 }\right\rfloor=\left\lfloor m+\frac{1}{2}\right\rfloor=m$


$a^{log_b n}$


$M = \begin{pmatrix} a^2 & a & 1 \\ b^2 & b & 1 \\ c^2 & c & 1 \end{pmatrix}$


$a_n = an^{log_b(n)}$


$ S = \frac{1}{2} \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \left(\frac{m}{n} + \frac{n}{m} \right) \frac{(-1)^{m+n}}{m^{2} + n^{2}} = \frac{1}{2} \left( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \right)^{2} = \frac{1}{2}\log^{2} 2. $


$\begin{align*}\int_0^\infty\dfrac{\sin^5x}xdx&=\dfrac1{16}\left[\int_0^\infty \dfrac{\sin5x}xdx+20\int_0^\infty\dfrac{\sin^3x}xdx - 5\int_0^\infty\dfrac{\sin x}xdx\right]\\&=\dfrac1{16}\left[\dfrac \pi 2 + 5\pi - \dfrac{5\pi}2\right]\\&=\dfrac{3\pi}{16}\end{align*}$


$\dot{x}_{ag}=\left[\matrix{\dot{x}\\ \dot{f}}\right]=\left[\matrix{Ax+Bu\\ 0}\right]=\left[\matrix{A & 0\\ 0 & 0 }\right]x_{ag}+\left[\matrix{B\\ 0}\right]u$


$\lim_{x \to c} |f(x)| = L$


$\int_{0}^\pi \frac{\sin(nx)}{\sin x} dx$


$z=\sqrt[4]{2}e^{\frac{\pi i}{6}}$


$C=\mathbb{R}^{n+1} -\mathbb{R}^{n-1}$


$1+2+3+4\dots = A-\frac{1}{12}$


$a\in V_a\subseteq \overline{V_a}\subseteq U_a$


$ n! = \sqrt{2\pi n} \left(\frac{n}{e}\right)^n e^{\Theta(1/n)} $


$\lim_{x\rightarrow 2^-}f(x)=\lim_{x\rightarrow 2^+}f(x)=\lim_{x\rightarrow 2}f(x)=3,$


$\mathbb R^{n+1} = \mathbb R^n\times\mathbb R$


$X=\frac{c(1+r)^{T+1}+(1-c)r^2}{r(1+r)((c-r)(1+r)^{T}+(1-c)r)}$


$F(x) = \int_a^x f(t) dt, a \le x \le b$


$Y Y = Y (Y Y ).$


$||A||_2 <= \sqrt{n}||A||_\infty$


$ \sum_{n\geq 1}\frac{1}{n^{2+\cos n}}, \qquad \sum_{n\geq 1}\frac{1}{n^{2-\cos n}} $


$d(p^{1-\gamma} T^\gamma)=(1-\gamma) p^{-\gamma} T^\gamma dp + \gamma p^{1-\gamma} T^{\gamma-1} dT$


$\sin^2\varphi + \cos^2\varphi = 1$


$ \frac{Y(y)-Y''(y)}{Y(y)}$


$\frac{\sqrt{1+y}}{1+\sqrt{1+y}}=\frac{\sqrt{1-y}}{1-\sqrt{1-y}}$


$f_n=\left(nf_1+(n-1)f_0\right)(-1)^{n-1}$


$\phi(x)=\frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}}$


$x = (p_1,p_2,p_3,\dots,p_n)$


$\|A\|_2^2 = Tr(A^TA)$


$\sigma_i(A) = \sqrt{\lambda_i (A^T A)}$


$\sum\limits_{r=1}^{n}\frac{1}{r(r+1)(r+2)}=?$


$\left\lfloor \frac{x}{n}\right\rfloor = \left\lfloor \frac{\lfloor x\rfloor}{n}\right\rfloor$


$x+1=\frac {a+b}b$


$\int_{0}^{1} \frac{1}{(1+x^2)^n} dx$


$[x,y]=[h,k]g$


$ \int \text{d} \Omega_{1} \text{d} \Omega_{2} f(\gamma) = 8 \pi^{2} \int \sin \gamma \, \text{d}\gamma f(\gamma).  $


$\alpha\omega=\alpha$


$\chi(n) = \left(\frac{n}{d}\right)$


$ \begin{align} \int_0^\pi\frac{\sin^n(x)}{x}\,\mathrm{d}x &\le\int_0^\pi\left(\frac{4x(\pi-x)}{\pi^2}\right)^n\frac{\mathrm{d}x}x\\ &=4^n\int_0^1x^{n-1}(1-x)^n\,\mathrm{d}x\\ &=4^n\frac{\Gamma(n)\Gamma(n+1)}{\Gamma(2n+1)}\\ &=\frac{4^n}{n\binom{2n}{n}}\\ &=2\prod_{k=1}^{n-1}\frac{k}{k+\frac12}\\ &\le2\prod_{k=1}^{n-1}\sqrt{\frac{k}{k+1}}\\ &=\frac2{\sqrt{n}} \end{align} $


$d(\gamma(0), \gamma(s)) + d(\gamma(s), \gamma(t))+d(\gamma(t), \gamma(1))\le \left(s + (t-s) + (1-t)\right)\, d(\gamma(0), \gamma(1)) = d(\gamma(0), \gamma(1))$


$S = \{w^0, w^1, w^2, ....., w^n\}$


$\sum_{i=1}^n \|e_i-v_i\|^2 < 1$


$\left\{ \matrix{   \sin \phi  = {{{1 \over {\sqrt 3 }}} \over {\sqrt {{1 \over 3} + 1} }} = {1 \over 2} \hfill \cr    \cos \phi  = {1 \over {\sqrt {{1 \over 3} + 1} }} = {{\sqrt 3 } \over 2} \hfill \cr}  \right.\,\,\,\,\, \to \,\,\,\,\,\phi  = {\pi  \over 6}$


$ 1/|x|^{n-\alpha} $


$ \operatorname{id}^{*} \circ f = \left( f^{*} \circ \operatorname{id} \right)^{*} = \left( f^{*} \right)^{*} = f, \\ f \circ \operatorname{id}^{*} = \left( \operatorname{id} \circ f^{*} \right)^{*} = \left (f^{*} \right)^{*} = f $


$\sum_{n=-\infty}^{-1} + \sum_{n=1}^{\infty}$


$\sum_{N=2}^\infty Np_N=\sum_{N=2}^\infty\frac{N(N-1)}{N!} =\sum_{N=2}^\infty\frac1{(N-2)!}=\cdots.$


$\lim_{n \to \infty}\frac{1}{n} \sum_{r=1}^n f\left ({r\over n} \right) $


$\sum \chi_{I_j}\le 2$


$A_0\subseteq A_1\subseteq \cdot \cdot \cdot \subseteq A_n \subseteq \cdot \cdot \cdot $


$\lim_{n \rightarrow \infty} f(x_n)= \sup_{x<m} f(x).$


$V=S\oplus T$


$\int\frac{1}{(x^2+x+1)^{1/2}} dx$


$\omega+\alpha=\alpha$


$ \frac{\;\;\frac{a^{3/2}}{b^3}\;\;}{\frac{a^{-1}}{b^2}} $


$(x+a_n)(n-1)(x+a)^{n-2} =(n-1)(x+a)^{n-1} $


$\lim_{x\rightarrow2^+}f'(x) = \lim_{x\rightarrow2^-}f'(x)$


$\frac{2\sin(a)+\sec(a)}{1+\tan(a)}=\frac{1+\tan(a)}{\sec(a)}$


$K \subset D\subset \bar{D} \subset A$


$f_X (x) = e^{-x/\theta}/\theta$


$b_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx) dx.$


$(t^∗)^{\mathcal A}(a) = f_n^{\mathcal A}(s_1(x/ \tau)^{\mathcal A}(a), \ldots,  s_n(x/ \tau)^{\mathcal A}(a))$


$=\left[\begin{array}{ccc|c}1&-2&2&0\\2&k&-1&3\\1&-1&3&-5\end{array}\right]$


$ \sum_{k=1}^n \cos \frac{2 \pi k}{n} = 0 $


$\displaystyle y=\frac{bx}{x-a}$


$ \begin{pmatrix} 2 & 2 & -5 & 2 \\ 1 & 1 & -2 & 1 \\ 1 & 0 & -3 & 4  \end{pmatrix}  $


$[f] \in \mathbb{R}[x] / (x^2 +1)$


$\overline{\mathbb R},~$


$\sum\limits_{n=0}^{\infty}\sum\limits_{r=0}^{n}\left(\frac{1}{(n-r)!}a^{n-r}\right)\left(\frac{1}{r!}b^{r}\right)$


$t f(p) + (1-t) f(p')$


$\sum_{n=1}^{\infty} (-1)^n=\sum_{n=1}^{\infty} (-1)^{2n}-\sum_{n=1}^{\infty} (-1)^{2n+1} ``=" \infty-\infty.$


$\text{re } \gamma_k = \text{im } \gamma_k = 0$


$n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}$


$^+e$


$\lim_{n\to\infty}{\dfrac{1}{n}\sum_{k=1}^{n}{\left(\dfrac{n}{k}-\left\lfloor\dfrac{n}{k}\right\rfloor\right)}}$


$\lim_{n \to \infty} \inf t_n = t_*$


$1-(P(A^c)+P(B^c))=1-((1-P(A))+(1-P(B)))$


$M = F \oplus T$


$\lim_{n \to \infty} \frac{1}{\sqrt{n}} \sum_{k=1}^n \frac{1}{\sqrt{k}}$


$\lim\limits_{n\to \infty} \sqrt{x_n} = \sqrt{\lim\limits_{n\to \infty} x_n}$


$ \det M=(k-\lambda)^{n-1}(\lambda(n-1)+k). $


$n! \sim \sqrt{2\pi n}\cdot (\frac{n}{e})^n$


$\{a,b,a+b,a-b\}$


$E[(X+Y)^p] \le 2^p (E[X^p] + E[Y^p])$


$f_n(x)=\frac{\sin x\sin nx}{x^2}$


$\sum_{n=1}^{\infty} \frac{1}{n^2+2^n}=\sum_{n=1}^{k} \frac{1}{n^2+2^n}+\sum_{n=k+1}^{\infty} \frac{1}{n^2+2^n}$


$x_n < \frac{n(n+1)}{2} $


$ M=\left (\begin{array}{cc} A &-B\\ B & A \end{array} \right ) $


$\det{A^T}=\sum_{\tau\in\Sigma_n}\epsilon(\tau)a_{\tau(1),1}\cdots  a_{\tau(n),n}=\det{A}$


$e_i,\; m+1\le i\le n$


$[x, y] = \mathscr{C}(a, b)$


$C_1\subseteq V\subseteq \overline{V}\subseteq C_2^c$


$F=S+T$


$\frac{ \alpha\beta}{\gamma} = \frac{\alpha\beta\gamma}{\gamma^2} = - \frac{q}{\gamma^2} $


$\frac{d}{dx}(x-r)^mq(x)=(x-r)^{m-1}(mq(x)+(x-r)q'(x))$


$\|u\|_p = \frac{|\langle u,v\rangle |}{\|v\|_q}$


$d(x,u)=d(y,v)$


$\frac{f}{||f||+||g||}$


$(1,1,-1,1,f(1,1,-1,1)) \in \mathbb{R}^5$


$\det \begin{pmatrix} x^2 & y^2 & z^2 \\ x & y & z \\ 1 & 1 & 1 \end{pmatrix} = (x-y)(x-z)(y-z)$


$F(x) = \frac{1}{2}\Big(\frac{1}{(1-x)^3} - \frac{1}{(1-x)(1-x^2)}\Big) = \frac{(1-x^2) - (1-x)^2}{2(1-x^2)(1-x)^3} = \frac{x}{(1-x^2)(1-x)^2}$


$\left\{ {\matrix{    {{a_n},\mathbb{N}_{even}}  \cr     {0,\mathbb{N}_{odd}}  \cr   } } \right.$


$b_j(x) = (f(x) - \frac{1}{|I_j|} \int_{I_j} f(y) dy) \chi_{I_j}(x)$


$P(n-3)\implies P(n)$


$B = \{ (x,y) \in \mathbb{R}^2 |   y \leq x \leq y^2+1,  0 \leq y \leq 1 \}$


$\phi: \mathbb{Z}/N\mathbb{Z}\rightarrow E[N](S)$


$\forall~\epsilon >0,~\exists~\delta > 0 :~ |x - a|<\delta \implies |fx - L|<\epsilon$


$I=A_1\times...\times A_m$


$\begin{align}&\quad 2^{k+1} \\[1ex]&= 2(2^k) \\[1ex]&\geq 2(k+1) \tag 1 \\[1ex]&= (k+1)+(k+1)\\[1ex]&\geq (k+1)+1 \tag 2\end{align}$


$T_x X = T_x U$


$S = 1 + 2+ 4 + 8 + 16 + \cdots$


$\lim_{n\to\infty} ((1+\frac{1}{n^2})(1+\frac{2}{n^2})(1+\frac{3}{n^2})\cdots(1+\frac {n}{n^2}))$


$x= \frac{1}{4}, y= \frac{1}{4}$


$\frac{a+b}{b}=1+\frac{a}{b}=1+\frac{1}{\frac{b}{a}}=1+\frac{1}{\frac{a+b}{b}}$


$ \mathcal G_1\supset\mathcal G_2\supset\cdots\supset\mathcal G_n\supset\mathcal G_{n+1}\supset\cdots $


$ \frac{(1-p)^x\cdot p}{1-(1-p)}=(1-p)^x.$


$\lim_{n\to \infty}\mu(A_n)=1$


$\mathcal{O}\subset \mathbb{R}^n\times \mathbb{R}\rightarrow \mathbb{R}^n$


$\langle X_i : i < \alpha^{+} \rangle$


$ M_\gamma = \begin{pmatrix}\cos(\gamma) & -\sin(\gamma)\\ \sin(\gamma) & \cos(\gamma)\end{pmatrix}, \qquad \qquad \gamma  = m \pi/2 $


$\sqrt{z} = \sqrt{|z|} \exp(i\frac{\varphi}{2})$


$ \lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!} $


$\sum\limits_{N\geq 1} \frac{N}{N^2} = \sum\limits_{N\geq 1} \frac{1}{N}=\infty$


$\forall \epsilon \gt 0, \exists \delta \gt 0 ~~s.t.~~ |x-2| \lt \delta ~~implies~~ |\alpha(x) - \alpha(2)| = |x-2| < \epsilon  $


$k\cong (k[X]/(X^n))/(X)$


$ \zeta(1-s) = \frac{1}{\pi} (2 \pi)^{1-s} \cos(\frac{\pi s}{2}) \Gamma(s) \zeta(s) $


$c_1(\mathcal T_X)=c_1(TX) = c_1(T^{1,0}X \oplus T^{0,1}X) = c_1(T^{1,0}X)+c_1(T^{0,1}X)$


$\exists \varepsilon > 0, \exists \delta > 0 : |x-a| < \delta \Rightarrow |f(x)-f(a)| \geq \varepsilon$


$\int_0^\infty f(x)\ dx=\infty$


$\mathfrak{A} = \sum_{n=-\infty}^{\infty} 2^{-n^{2}}$


$ D_3~=~cov(x,y) $


$\lim_{x\to \alpha^+} f(x) \not\exists$


$20!+18, 20!+19, \cdots, 20!+23$


$|y - x_{n+1}| < \epsilon/2$


$1-2\left ( \frac{1}{2}^{n-1} \right )$


$\left \lfloor \dfrac{\left \lfloor en! \right \rfloor}{n} \right  \rfloor=\left \lfloor e(n-1)! \right \rfloor$


$\sum_{n\geq1}\frac{1}{2n^{2}}+\sum_{n\geq1}\left(\frac{1}{2\left(4+n^{2}\right)}-\frac{4}{\left(4+n^{2}\right)^{2}}\right)$


$ c_n = 3+\frac{n(n+1)}{2} = \frac{n(n+1) + 6}{2} = \frac{n^2 +n + 6}{2} $


$\inf\{F_\gamma(\gamma)\mid\gamma<\alpha\}\le \inf\{F_\gamma(\beta)\mid\beta\le\gamma<\alpha\}=0$


$\gamma,\dot\gamma,\ddot\gamma$


$\mathrm{Spec}(\mathbf{C}[X]/(X^2+1))=\mathrm{Spec}(\mathbf{C})\coprod\mathrm{Spec}(\mathbf{C})$


$n{n-1 \choose 2}={n \choose 2}{(n-2)}$


$x-3\mid x^3-3$


$ \begin{align} \int_0^\infty\frac{\sin^2(x)}{x}\,\mathrm{d}x &=\sum_{k=1}^\infty\int_{(k-1)\pi}^{k\pi}\frac{\sin^2(x)}{x}\,\mathrm{d}x\\ &\ge\sum_{k=1}^\infty\frac1{k\pi}\int_{(k-1)\pi}^{k\pi}\sin^2(x)\,\mathrm{d}x\\ &=\sum_{k=1}^\infty\frac1{2k} \end{align} $


$\lim \limits_{n \to \infty} (\sum \limits_{r=1}^{n^2} \frac{1}{(n+r)a_{n,r}}-\ln n)= - \infty$


$\frac{(1+r)^n - 1}{(1+r)^n \times r}=y=8.491875$


$n!+2, n!+3, \ldots, n!+n$


$ \color{blue}{\sqrt{1+\frac{d^2y}{dx^2}}=x+\frac{dy}{dx}}\implies 1+\frac{d^2y}{dx^2}=\bigg[x+\frac{dy}{dx}\bigg]^2 $


$\omega+1 = \sqrt{3}e^{\frac{i\pi}{6}}$


$xRy\implies yRx,\quad$


$v_1^{1,1}, v_1^{1,2}, v_2^{2,1}$


$x = t^{1/2n}$


$P(n) \implies P(f(n))$


$n! = 1+\sum\limits_{r=1}^{n-1}( r\cdot r!)$


$e^x = \lim \left({1 + \frac x n}\right)^n$


$\left| {\begin{array}{*{20}{c}} x&y& \cdots &y\\ y&x& \cdots &y\\  \vdots &y& \ddots &y\\ y&y&y&x \end{array}} \right| = {(x - y)^{n - 1}}\left[ {x + (n - 1)y} \right]$


$A_1\subseteq A_2\subseteq A_3\subseteq A_4 ...\subseteq A_n$


$ \left[     \begin{array}{ccc|c}       1&1&k&6\\       1&k&1&3\\       k&1&1&7     \end{array} \right] $


$x^9 + x^6 + x^4 - x^2 + 1=0$


$Cov(X,Y) > 0$


$E|X|^r < \infty$


$\phi(x) = \sum_{k=1}^N a_k \chi_{A_k}(x)$


$\gamma(s)=\gamma(0) + \int_0^s C\,ds = \gamma(0) + Cs$


$P(X_{n+1}=y | X_n=x)=P(X_1=y | X_0=x)$


$x^6+3 x^5+6 x^4+3 x^3+9 x+9.$


$F=P\oplus P'$


$ p_1p_2\mid n\iff p_1\mid n\text{ and }p_2\mid n\tag{3} $


$\varphi^+$


$\tan(a)=\frac{\sin(a)}{\cos(a)}=\frac{\sin(a)}{\sqrt{1-\sin^2(a)}}$


$20 + 10 + 10 = 40$


$|u_n (x)|\leq \frac {2|x|}{n^2+x^2}\leq \frac{2|x|}{n^2} .$


$A_0, A_1, ..., A_n, A_{n+1}$


$R e^{\frac{i\pi}{n}}$


$f_{x_{max},x_{max-1}}(x,y)=\frac{N!}{(N-2)!}e^{-x-y}(1-e^{-y})^{N-2}.$


$ p(M) \leq \lambda(M) \leq \pi(M) $


$\implies (1+\frac {x}{n})^n\le e^x $


$S = \{S_1, S_2, S_3, \dots, S_n\}$


$\,\frac{\pi}{\sin(\pi s)}+\Gamma(1-s)\,\zeta(1-s)\,$


$\sum_{n=1}^\infty\frac1{n^2}=\sum_{n=1}^{a-1}\frac1{n^2}+\underbrace{\int_a^\infty\frac1{x^2}~\mathrm dx}_{=1/a}+\frac1{2a^2}+\sum_{k=1}^p\frac{B_{2k}}{a^{2k+1}}+R_p$


$\text{P}(A) = 1/6$


$(e_1, e_2, e_3, \ldots)$


$(a+b,a-b)=(15,1)\; \text {or} \;(5,3)$


$\frac{(1+x^2)^{n+1} - 1}{x^2(1+x^2)^{n+1}}$


$2^kp_1p_2\ldots p_s$


$[x,y]=rx+sy$


$gx-x,x\in M,g\in G$


$\sum_{n\leq x}\frac{\varphi(n)}{n^{2}}=\sum_{n\leq x}\frac{1}{n^{2}}\sum_{d\mid n}\mu(d)\frac{n}{d}=\sum_{{q,d}\atop{qd\leq x}}\frac{\mu(d)}{d^{2}q}=\sum_{d\leq x}\frac{\mu(d)}{d^{2}}\sum_{q\leq\frac{x}{d}}\frac{1}{q} \tag{4}$


$x^3-x\ge x^3 -x$


$1+2+3+4+\dots = -1/12.$


$f_i(x)=\frac{\langle x, v_i \rangle}{\|v_i\|^2}$


$(f'(f^{-1}(x)))'=f''(f^{-1}(x))·(f^{-1}(x))'$


$=\lim_{h\to0^+}\frac{(1+2h)^2-(1+6h-h^2)}{h(1+2h+\sqrt{1+6h-h^2})}$


$p^r - p^{r-1}$


$D=\{ (x,y) \in \mathbb{R}^2: a\leq x\leq b, c\leq y \leq d   \}$


$\forall m \leq n \ P(m) \implies P(n')$


$\frac{1}{2^n}\rightarrow \frac{1}{2^{n-1}}$


$ (\gamma'')^\perp = \langle \gamma'' , \gamma\rangle \gamma + \langle \gamma'', \nu\rangle \nu.   $


$f(xy)=f(f(x)+f(y))$


$S=\{\alpha f+ \beta g + \gamma h: |\alpha|\le a, |\beta|\le b, |\gamma|\le c\}$


$ax + my = d$


$\int_k^{k+1}f(\frac{u}{n})g(u)du=\int_0^1f(\frac{t+k}{n})g(t+k)dt=\int_0^1f(\frac{t+k}{n})g(t)dt$


$|\nabla(u^2)|^2=2u^2|\nabla u|^2$


$\Big\lfloor\sum_{r=1}^N\log_br\Big\rfloor+1.$


$\sqrt[n]{x_1 x_2 \cdots x_n} \leq \frac{x_1 +x_2 + \cdots + x_n}{n} $


$H = \{(x,y)| a \le x \le b; c \le y \le d\}$


$\dfrac {\sqrt3 - \sqrt2}{\sqrt3 - \sqrt2}$


$L_2 = \{ r \in \mathbb{Q} : 0 < r < 2 \} $


$Cov(X,Y)\neq 0.$


$\mathbb E[X^2]\geqslant \mathbb E[|X|]^2. $


$ \left\lfloor\frac{\left\lfloor\frac ac\right\rfloor}b\right\rfloor=  \left\lfloor\frac{qb+q''}c\right\rfloor=q.$


$G:\mathbb R^{n+1}\to\mathbb R^n$


$\mathbb{C}^{+}$


$\therefore \int_0^\pi f^2(x) \, dx = \int_0^\pi f(x) \left(\sum_{n=0}^\infty \cos(nx)\right) \, dx = \sum_{n=0}^\infty \int_0^\pi f(x) \cos(nx) \, dx = 0$


$\lim_{k \to \infty}|\frac{(k+1)^3}{k^3}| = |\frac{{(k(1+\frac{1}{k}))}^3}{k^3}| = |\frac{k^3(1+\frac{1}{k})^3}{k^3}| = 1$


$ \int_{0}^{1}\frac{\log^2 (x+1)}{x}$


$\begin{align} \int_0^\infty f_X(x)\,dx&=\frac{4c}{3}\Gamma(4)=8c \end{align}$


$ w(x) = 2 \sqrt{\frac{x}{c}}$


$\lim_{n\to\infty} \mu(A_n)=\mu(\bigcup_{n} A_n)$


$P(p) \implies P(p+1)$


$m_r \equiv \mathbb{E}|X|^r$


$|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$


$ f (xf (y)) f (y)=f (x+y) $


$A=\int_{0}^{\pi }\frac{xsinx}{1+sin^2x}dx =\int_{0}^{\pi }\frac{(\pi-x)sin(\pi-x)}{1+sin^2(\pi-x)}dx\\\int_{0}^{\pi }\frac{(\pi-x)sin(\pi-x)}{1+sin^2(\pi-x)}dx\\=\int_{0}^{\pi }\frac{(\pi-x)sin(x)}{1+sin^2(x)}dx\\=\int_{0}^{\pi }\frac{(\pi)sin(x)}{1+sin^2(x)}dx+\int_{0}^{\pi }\frac{(-x)sin(x)}{1+sin^2(x)}dx\\A=\int_{0}^{\pi }\frac{(\pi)sin(x)}{1+sin^2(x)}dx-A\\$


$f(xf(y))=f(x)y$


$ \begin{align} \frac{(1+x)^{n+1}-(1+y)^{n+1}}{x-y} &=\sum_{k=0}^{n+1}\binom{n+1}{k}\frac{x^k-y^k}{x-y}\\ &=\sum_{k=0}^{n+1}\binom{n+1}{k}\sum_{j=1}^kx^{j-1}y^{k-j}\tag{7} \end{align} $


$f(xf(y)+f(x))=yf(x)+x$


$P(AB) = 1/6,$


$ \frac{1-\dfrac{b^{n}}{a^{n}}}{1-\dfrac{b}{a}} =\frac{\dfrac{a^n-b^n}{a^n}}{\dfrac{a-b}{a}}= \frac{a^n-b^n}{a^n}\frac{a}{a-b}= \frac{1}{a^{n-1}}\frac{a^n-b^n}{a-b} $


$ \Bigl|\int_0^x\sin (t^2)\,dt\Bigr|\leq\int_0^x|\sin(t^2)|\,dt $


$xy \vee xz \vee yz$


$a_n =\int_{0}^{1}  \frac{x^n \sin(\pi x)}{1-x} \, \mathrm{d}x$


$d(x,z) < d(x,y) + d(y,z)$


$f(a+b)=f(a) \circ f(b)$


$f \colon K[T_1^{\pm 1}, \dots, T_r^{\pm 1}] \to \mathcal O_X(U)$


$\lambda<\alpha^+$


$\int_{-\pi}^{\pi}\dfrac{\sin nx}{(1+\pi^{x})\sin x}dx$


$\Pr[X\ge x]=0$


$x\in V\subseteq U$


$ 0 \leq \varepsilon\cdot \mathbb{P}\{ \lvert X - X_n\rvert \geq \varepsilon\} \leq \mathbb{E} \lvert X - X_n\rvert. $


$f(x)=\frac { x }{ { e }^{ x }+1 } $


$x_{1},x_{2},x_{3}\in \Bbb{N}\setminus\{0\}$


$c=\{\{\emptyset\}\}$


$n! \ge \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n$


$\gcd(nn!, n!+1)=1$


$\left|\int_\gamma f(z)\right|\le\int_\gamma |f(z)|\le M\int_\gamma |z^n|\le M'|z|^{n+1}$


$xRy \implies yRy$


$X=\left\{x_{1},x_{2},x_{3},.....,x_{N}\right\}$


$m = p_1 p_2 \cdots p_k$


$M = F \oplus T(M)$


$\int 2\pi r ds$


$\mathbb{E}|X - m|^3 \leq \mathbb{E}|X|^3 (1 + \frac{m}{\sigma})^3$


$x/4 = \tan \theta$


$s_{n}(x) = \sum \limits_{i = 1}^{m} \alpha_{i}*\chi_{A_{i}}(x)$


$\sqrt{1+\left(\frac{df}{dt}\right)^2} = \operatorname{const.}$


$\{x\}=f^{-1}(f(\{x\}))=f^{-1}(\{f(x)\})=f^{-1}(\{f(x')\})=f^{-1}(f(\{x'\}))=\{x'\}$


$n! + 2, n! + 3, n! + 4... n! + n$


$Var=|t-s|$


$P[X_t=x|T=t]= \pi(x)$


$P = \frac {k^3 - (k-1)^3}{216}$


$f_n(x)= \dfrac{\sin 2 \pi x \,\sin 2\pi n x}{x^2}$


$ A+iB\mapsto \begin{pmatrix} A & -B \cr B & A \end{pmatrix}. $


$A = \{a_1,a_2,a_3,\ldots,a_n\}$


$\sum_{n=0}^\infty \frac{1}{2^{n+2}}$


$A_1\subseteq A_2\subseteq A_3\subseteq\dots $


$\lim_{n\to\infty} \mu(A_n)=\mu(X).$


$x! \sim \left(\frac{x}{e}\right)^x\sqrt{2\pi{x}}\left(1 + \frac{1}{12x}\right)$


$P(X\mid A_1 \cap A_2 \cap \cdots \cap A_n \cap A_{n+1})$


$(P ∨ ¬P) = (F ∨ T) = T$


$f\big(xy + f(y)\big) = y\,f(x)$


$\left(1+\frac {x^2}n\right)^n\ge \left(1+\frac{x^2}{2}\right)^2$


$x_{\gamma}  = \frac{1}{1+\gamma}x_{\gamma=0}.$


$ z \mapsto \vartheta(z + \frac{1+\tau}{2};\tau)$


$|1/x| = 1/|x|$


$\gamma=\gamma(t)=(\gamma_1(t),...,\gamma_n(t))$


$f_X(x) = \frac{2(\theta -x)}{\theta^2}, \ \ \ \ 0<x<\theta $


$\sum_{\substack{{k_1,k_2,\dots, k_n\geq0}\\{k_1+2k_2+3k_3+\cdots+nk_n=n} }}\frac{(k_1+k_2+\dots+k_n)!}{k_1!k_2!\dots k_n!} = 2^{n-1}.$


$a^{({}^2a)^b*({}^2b)^a}*b^{({}^2b)^a*({}^2a)^b}$


$f(ab)=f(a)+f(b), f(1)=0$


$d(x,y)\leqslant d(x,x)+d(y,x)$


$a^{log_ax} = x$


$\int\sum_{n=-\infty}^\infty e^{-\pi{n^2}/ x}  dx $


$\Delta = \begin{vmatrix} x & y & z \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix}$


$A=Q\sqrt{A^TA}$


$xRy\wedge yRx\wedge x\neq y$


$0 = d^2 = (\partial + \bar\partial)(\partial +\bar \partial) = \partial ^2 + (\partial \bar\partial + \bar\partial \partial) + \bar\partial^2,$


$ \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} = \color{red}{\sum_{n\geq 1}\frac{3}{n^2\binom{2n}{n}}}\tag{A}$


$F(x) = \int_a^x f(t)\,dt$


$\sum_{k=1}^{n-1}  {\cos {\frac{2k\pi}{n}}} = -1$


$[x,y]^{\alpha}=[x,y]$


$P+H=F$


$P(M) = 1/6$


$\lfloor \frac{a}{c} \rfloor+\lfloor \frac{b}{c} \rfloor \le \lfloor \frac{a+b}{c} \rfloor $


$\lim_{x \to c} f(x) = L,$


$0=\inf_{n \geq 1} \nu(E_n) = \lim_{n \to \infty} \nu(E_n).$


$\int_0^{+\infty} f(t) \sin(t) dt$


$\begin{cases} x = 0\\ x+y = 0\\ y = 0 \end{cases} \Rightarrow \begin{cases} x = 0\\ y = 0 \end{cases}$


$\int_0^\infty f(x)\,dx.$


$\dfrac{\tan^2\theta}{\tan\theta\sin\theta}-\dfrac{\sin^2\theta}{\tan\theta\sin\theta}\Rightarrow \dfrac{\tan^2\theta-\sin^2\theta}{\tan\theta\sin\theta}$


$R_{1}(x) = o((x - a)^{n - 1})$


$\frac{T_4}{T_2}=\frac {a+3d}{a+d}=\frac {\frac{a}{d}+3}{\frac{a}{d}+1}=\frac {r+3}{r+1}=\frac {\frac{1}{3}+3}{\frac{1}{3}+1}=\frac{5}{2}$


$ \tag{1} [T_a,T_b] = T_a $


$x^{k+1} - y^{k+1} = x^{k} + \Delta x^{k} - y^{k} - \Delta y^{k} \\ = F'(x^{k})^{-1}\left( F'(x^{k})(x^{k} - y^{k}) - \int_{0}^{1}F'(y^{k} + t(x^{k}-y^{k}))(x^{k}-y^{k})\text{d}t \right) + \\ \quad \quad F'(x^{k})^{-1}(F'(y^{k}) - F'(x^{k}))\Delta{y}^{k}.$


$\limsup_{n\to\infty} \mu_n(K) \leq \mu(K)$


$3^x + 4^x = 5^x$


$\begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}\neq0$


$f^2(x) = \dfrac{7x + 18}{3x + 10}$


$(RS+R)^* = (RS)^*R(RS)^*R(RS)^*....R(RS)^* = R^*(RS)R^*(RS)...R^*$


$P_2 = P \oplus P$


$\begin{align}   \int\limits_0^{\infty} f(x)\,\mathrm{d}x \;\;&=\;\;      \int\limits_0^1 f(x)\,\mathrm{d}x +      \int\limits_1^{\infty} f(u)\,\mathrm{d}u     \\\;\;&=\;\;\int\limits_0^1 f(x)\,\mathrm{d}x +      \int\limits_0^1 \frac{f\!\left(\frac{1}{x}\right)}{x^2}\,\mathrm{d}x \;\;=\;\;      \int\limits_0^1 \frac{x^2f(x)+f\!\left(\frac{1}{x}\right)}{x^2}\,\mathrm{d}x \end{align}$


$ \sum_{n=1}^{+\infty}\frac1{1+n}=\sum_{n=2}^{+\infty}\frac1{n}=+\infty $


$R = \mathbb{R}[x]/(x^n + 1)$


$ \begin{cases} x=39t\\ y=-36 t\\ z=20t \end{cases} $


$dL^2 = \left(1 + \left(\frac{dy}{dx}\right)^2\right) dx^2$


$=\log\left(\frac{\left(1+\tan\frac{x}{2}\right)^2}{\left(1-\tan\frac{x}{2}\right)\left(1+\tan\frac{x}{2}\right)}\right)$


$\left(1+\frac1n\right)^n\ge 2$


$C = \begin{bmatrix}cov(x,x)&cov(x,y)&cov(x,z)\\cov(y,x)&cov(y,y)&cov(y,z)\\cov(z,x)&cov(z,y)&cov(z,z)\end{bmatrix}$


$\mathbb{Cov}(X,Y) = 0$


$\int_{-\infty}^\infty \frac{x^{2n}}{(x^2 + 1)^{n + 1}}\ dx.$


$\lim\limits_{n\to \infty}\ \frac{1+\sqrt[2]{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}=1$


$x^3 - x^2 \in \mathbb{Z}$


$d(x,M)=\inf_{y\in M}{\|x-y\|}=\frac{|\langle f,x\rangle|}{\|f\|}.$


$A=B=C=D=\mathbb{R}$


$\dots\subseteq\overline{V_{i+1}}\subseteq\overline{V_i}\subseteq\dots$


$P(k-1)\implies P(k)$


$\sum_{i=1}^{1000}f\left(\frac{i}{1000}\right),\qquad f(x) = \frac{4^x}{4^x+2}$


$F(x) - F(a) = \int_a^x f(t) dt$


$F([x,y]) = x$


$ E(D^2L^2)=E(D^2)E(L^2) = [Var(D)+E(D)^2][Var(L)+E(L)^2] = Var(D)Var(L)+Var(D)E(L^2)+Var(L)E(D)^2+E(D)^2E(L)^2$


$1(1!)+\dots+n(n!) = (n+1)!-1$


$\frac{1}{2^{x - 2}} = 2^{2 - x}$


$\frac{1}{X} + \frac{1}{Y} = 7$


$\lim_\gamma \delta_\gamma = \bigcup_\gamma \delta_\gamma = \sup_\gamma \delta_\gamma$


$ M=\left( \begin{array}{cc} A&-\vec d^T \\ \vec c& b \\ \end{array} \right) $


$I_{i_0} \subseteq J_{j_0}$


$\sum\limits_{n=-\infty}^{\infty} (-1)^{n}e^{-\alpha n^2} = \sqrt{\frac{\pi}{\alpha}}\sum\limits_{n=-\infty}^{\infty} e^{-\frac{(2n+1)^2\pi^2}{4\alpha}}$


$f(a)+f(b)\not=f(a+b)$


$e^{x/n} > (1+\frac{x}{n})$


$2a, 2b,  a+b, a-b $


$T(\sum e_n) = \sum x_n$


$ G=\frac{X_1+X_2+...+X_n}{n}, $


$\{(x,y,z,w,l)\in \mathbb{R}^5 :x=y,z=w=l\}$


$\mathbf x = [x, y]$


$\lim_{n\to \infty}f(x_n) =\inf f(x)\geq -\infty$


$n! = (1 + o(1)) \sqrt{ 2 \pi n} \left( \frac{n}{e} \right)^n$


$\large{a^{log_a(c)} = c}$


$\gamma = \gamma(x) = \frac{1}{x^2}\tilde{\gamma}(x)$


$f(a+b) \leqslant f(a) + f(b)$


$ = \frac{(1 - \sin \varphi)^2}{ 1 - \sin^2 \varphi} = \frac{(1 - \sin \varphi)^2}{(1 - \sin \varphi)(1+\sin \varphi)} = \frac{1-\sin \varphi}{1+\sin\varphi}$


$f(x)={e^x\over e^x+1}, x\in \mathbb R$


$\dim A + \dim B = \kappa + 1 = \kappa = \kappa + 2 = \dim C + \dim D$


$F(x) = \int_a^x f(t)\,\mathrm{d}t.$


$z^n=w=se^{i\phi}$


$ \left.(\gamma\circ s^{-1})'\right|_{s(t)} = \left.(\gamma'\circ s^{-1})(s^{-1})'\right|_{s(t)} = \left.\frac{\gamma'\circ s^{-1}}{s'\circ s^{-1}}\right|_{s(t)} = \frac{\gamma'}{s'} = \frac{\text{d}\gamma/\text{d}t}{\text{d}s/\text{d}t} = \frac{\text{d}\gamma}{\text{d}s} $


$\frac{10/9}3 + 1 = \frac{1+3+9+27}{27}$


$x = \frac{- y^2 + 2ay - az}{a - z}.$


$(S^{n}-S^{n-1} )$


$M_2=\left(\begin{array}{} A & B\\ B^{H} &C \end{array}\right)$


$\langle\beta,m\rangle\in D$


$a_n= {n(n+3)(n-1) \above 1.5pt 2}$


$k=0 \rightarrow \left[\begin{array}{ccc}1&2&1\end{array}\right], k=1 \rightarrow \left[\begin{array}{ccc}1&-1&1\end{array}\right]$


$ P\left( \sup_{t \geq 0 } \exp\left\{a W_t - \frac12 a^2 t \right\} > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{1}{x} \quad\text{a.s.,}$


$\left  \lfloor \frac{a}{b} \right \rfloor + \left  \lfloor \frac{c}{d}  \right \rfloor =\left  \lfloor \frac{c+a}{b*d} \right \rfloor$


$\big[{T^{-1}\big]}^{\alpha}_{\beta} = (\big[{I\big]}^{\beta}_{\alpha})^{-1}\big[{T^{-1}\big]}^{\beta}_{\alpha}\big[{I\big]}^{\beta}_{\alpha}  =\big[{I\big]}^{\alpha}_{\beta}\big[{T^{-1}\big]}^{\beta}_{\alpha}\big[{I\big]}^{\beta}_{\alpha}$


$(1)\ \tan^2(\frac{1}{2}\theta) = \frac{\tan(\theta) - \sin(\theta)}{\tan(\theta)+\sin(\theta)}$


$\sum_{n=1}^{N}\frac{1}{2^{n-1}}\to 2$


$ \gamma(A) = \begin{cases} 0 & \text{if }A\text{ is countable}\\ 1 & \text{otherwise.} \end{cases} $


$\mathbb R^{N+1}\times \mathbb R^N$


$f(x)=\frac{a+x}{b+x}=1+\frac{a-b}{b+x}$


$\Rightarrow a=\lambda a \gamma\Rightarrow \lambda=\frac {1}{\gamma}$


$\cos(x,y)=\frac{\langle x,y\rangle}{||x||\cdot||y||}$


$P_n(x)=\int_{-x}^{1-x}f(x+t)Q_{n}(t)dt=\int_{0}^{1}f(t)Q_{n}(t-x)dt$


$[T_1, T_2]:=T_1\circ T_2-T_2\circ T_1$


$X=S\oplus T$


$\{ \dot \gamma, \ddot \gamma, \dots, \gamma^{(m)} \}$


$\sqrt{x}=\sqrt{r}e^{ \frac{i\theta}{2}}$


$\int_{-\frac{\pi}{2}}^{0} {\frac{1}{x\sin(x)}} dx+\int_{0}^{\frac{\pi}{2}} {\frac{1}{x\sin(x)}} dx$


$\gamma^* = \gamma([a,b])$


$L=\lim_{n\to\infty}\frac1{\ln n}\sum_{r=1}^{n^4}\frac1r,\qquad M=\lim_{n\to\infty}\left\lfloor\frac1{\ln n}\sum_{r=1}^{n^4}\frac1r\right\rfloor$


$\frac{(1+\sin(\theta)+i\cos(\theta))^2}{(1+\sin(\theta)-i\cos(\theta))(1+\sin(\theta)+i\sin(\theta))}$


$\pi \colon \mathbb{R}^{n+1} \rightarrow \mathbb{R}^n$


$\forall\epsilon \gt0 ,\, \exists\delta\gt 0 :|x-0|\lt\delta\Rightarrow|\frac{2+\sin x}{3-\cos x}x-0|\lt\epsilon$


$ \frac{\tan \theta \ + \ \cot \theta}{\sec \theta \ \cdot \ \sin \theta} \ = \ \frac{\tan \theta \ + \ \cot \theta}{\tan \theta } \ = \ 1 \ + \ \frac{\cos^2 \theta }{\sin^2 \theta } \ = \ \frac{\sin^2 \theta \ + \ \cos^2 \theta }{\sin^2 \theta } \ = \ \frac{1}{\sin^2 \theta } $


$|f '(z)| < |f(z)|$


$n! > \sqrt{2{\pi}n}(\frac{n}{e})^n$


$A_1\subset A_2\subset\cdots$


$ax+by=z$


$\lim_{x \to \ c-}f(x)=l$


$P(t) \implies P(t+1)$


$\alpha a + (\beta + \gamma + \delta) a' = (\alpha + \beta + \gamma + \delta)e\\ a'=\frac{(\alpha + \beta + \gamma + \delta)e - \alpha a}{\beta + \gamma + \delta}$


$ \left | F(\gamma(t)) \cdot \gamma'(t) \right | \leq \left | F(\gamma(t)) \right | \left | \gamma'(t) \right | $


$|f(z)| \leq |f'(z)|$


$\neg B\lor \neg\neg B$


$f(e^{-\gamma})=e^{e^{-\gamma}-\gamma}$


$\alpha = \alpha^+- \alpha^-$


$y = (y,y',y'')$


$u^+$


$\frac{4}{x}+\frac{7}{x-3}+\frac{1}{2}=1$


$\text{Im}(\gamma_1) = \text{Im}(\gamma_2)$


$(\tilde{x},\tilde{y},\tilde{z})=(1,1,1)$


$d(x,A) \le d(x,a) \le d(x,y) - d(y,a),$


$\left[\matrix{\bar x\cr \bar y\cr}\right]=\left[\matrix{x_0\cr y_0\cr}\right]+d\ \left[\matrix{\cos\phi&-\sin\phi \cr \sin\phi&\cos\phi\cr}\right]\>\left[\matrix{x\cr y\cr}\right]\ .$


$\neg \neg p \lor \neg \neg q$


$||x||=\sqrt{|\langle x,x\rangle|}$


$x[n] = \delta[n+1]+\delta[n]+\delta[n-4]$


$\gamma, \ddot \gamma \in \dot \gamma ^\perp$


$\gcd(a) = a$


$\frac{1}{\pi}\int_0^\pi |\cos(x)|dx=\frac{1}{\pi}\int_0^{\pi/2} \cos(x)dx + \frac{1}{\pi}\int_{\pi/2}^{\pi} -\cos(x)dx=\frac{1}{\pi}[\sin(\pi/2)-\sin(0)-\sin(\pi)+\sin(\pi/2)]=\frac{1}{\pi}(1-0-0+1)=\frac{2}{\pi}$


$ n! \approx \sqrt{2\pi n} \left( \frac{n}{e} \right)^n $


$\varepsilon = \frac{|\langle v,f \rangle|}{2 \| f \|_{L^2}}$


$a(n) = n^2 - n/2 + O(1)$


$\lim\limits_{n \rightarrow \infty} \frac{\sqrt{n+1} + 3}{\sqrt{n+2} - 4}$


$\lim\limits_{n\to \infty }\frac{1}{n}\sum\limits_{j=1}^na_j\:=\:\lim\limits_{n\to \infty }\left(a_n\right)$


$(x' + y') + z' = (x + y)' + z' = ((x+y) + z) = (x + (y+z))' = x' + (y+z)' = x' + (y' + z')$


$\nabla f=O(1/|x|)$


$A = \frac{R(1+r)^{-1}(1-(1+r)^{-n})}{1-(1+r)^{-1}} \; .$


$P(X=0) = \left(\frac12\right)^{n-2}$


$\psi = \frac{-xa\:+\:by}{a\:-x} \tag{11}$


$f_X(x) = \frac{2}{3}e^{-2x/3}$


$A_1\cap A_{n+1},\ldots,A_n\cap A_{n+1}$


$\int_0^{\infty} x f(x) $


$1-f,f\in M$


$\frac{dl}{d \beta} = \sqrt{r^2 + \frac{dr}{d \beta}^2}$


$[c_0, c_1]$


$ \lim_{n\to\infty} \frac{1}{\log n}\sum_{m=1}^n \frac{1}{m} = 1. $


$y=\frac{a+bx} {c+dx}+d$


$I'(a)=\int_0^{\pi/2}\frac{\ln(\tan x)}{\left(1+\tan^ax\right)^2}dx$


$\begin{cases}x=2t+1\\y=t\\z=3t-1\end{cases}$


$\sum^\infty_{-\infty} |\hat{f}(n)|^2 < \infty$


$\sqrt{1+\frac{\partial y}{\partial x}^2}$


$(\gamma', \gamma \times \gamma', \gamma)$


$\sum_{n=1}^\infty=\infty $


$ \begin{align} &\small n\frac{\frac1{n-1}-\frac1n}{(n-1)^{\frac{1-k}k}+(n-1)^{\frac{2-k}k}n^{-\frac1k}+(n-1)^{\frac{3-k}k}n^{-\frac2k}+\cdots+(n-1)^{-\frac2k}n^{\frac{3-k}k}+(n-1)^{-\frac1k}n^{\frac{2-k}k}+n^{\frac{1-k}k}}\\[9pt] &\le\frac{\frac1{n-1}}{kn^{\frac{1-k}k}}\\[6pt] &=\frac{n^{-\frac1k}}k\frac n{n-1}\\[12pt] &\to0 \end{align} $


$f(x) = \displaystyle\frac{2e^{2x}}{x^2}$


$ \left(\left(\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{    x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}\right)_{\left(\left(x_x^x\right)_{x_x^x}^    {x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x    }}}^{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^    x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}}\right)_{\left(\left(\left(x_x^x\right)_{x_x^    x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_    x^x}}\right)_{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x    ^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}}^{\left(\left(x_x^x\right)_{x_x^x}^{    x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}    }}}^{\left(\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}    ^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}\right)_{\left(\left(x_x^x\right)_{x_x^x    }^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x    ^x}}}^{\left(\left(x_x^x\right)_{x_x^x}^{x_x^x}\right)_{\left(x_x^x\right)_{x_x^x}^{x_    x^x}}^{\left(x_x^x\right)_{x_x^x}^{x_x^x}}}} $


$3 \mid a^3-a$


$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}a_k=\lim_{n\rightarrow\infty}\sum_{k=1}^{n}k=\lim_{n\rightarrow\infty}\frac{n(n+1)}{2}\rightarrow\infty$


$\pi:\mathbb{R}^{n+1}\to \mathbb{P}^n$


$[x,y]=x\overline{y}$


$\lim_{n\to\infty}n\sum_{r=1}^n\frac1{(n+r)^2}=\lim_{n\to\infty}\frac1n\sum_{r=1}^n\frac1{\left(1+\dfrac rn\right)^2} $


$x^2,x^3\in A$


$\infty > \mu(A) \geq \mu(A \cap B) = \mu(A)\mu(B) = \infty$


$\displaystyle \int_0^\infty \dfrac{\sin^5x}xdx=\dfrac{3\pi}{16}$


$<  \gamma+\beta\cup \lbrace  \gamma+\beta\rbrace$


$\log_a(1) = 0, \log_a(b) = 1 + \log_a(b/a)$


$0 = z = 1 + 2λ$


$\tan \theta = \frac{\sqrt 5}{2};$


$\langle\gamma'',  \nu\rangle = h(\gamma',\gamma')$


$d(x,z) \leq d(x,y) + d(y,z);$


$c_0 + c_1x$


$\bar{X}_n = {X_1 + X_2 + \cdots + X_n \over n}$


$\lim\limits_{n\to \infty} \frac{1+2\sqrt2+3\sqrt3+\ldots+n\sqrt n}{n^2 \sqrt{n}}$


$\frac1n\sum_{i=1}^nx_i=\frac{x_1+x_2+\ldots+x_n}n\;.$


$|f(z)| \le 2 |z|^2$


$a\in \Bbb F$


$\frac{\partial l(\theta, X)}{\partial \theta}\frac{\partial l(\theta, X)}{\partial \theta}^T$


$\left| \sum_{k=1}^n c_k - \sum_{n=1} ^\infty a_n \sum_{m=1}^\infty b_m\right| \le 2\left( \sum_{n=1}^\infty |a_n| + \sum_{m=1}^\infty |b_m| \right)\epsilon$


$\int_{0}^{\infty} \frac{\sin^{4}(x) }{x^{2}} \, dx $


$f(f(x)+f(y))=af(x)+f(a)+f(x)-f(y)x$


$\det \begin{pmatrix} A & B \\ B & A \end{pmatrix} = \det{(A+B)}\det{(A-B)}$


$\beta A \alpha^{1-\gamma}=((\beta A \alpha^{1-\gamma})^{1/\gamma})^\gamma=(\beta^{1/\gamma}A^{1/\gamma}\alpha^{(1/\gamma)-1})^\gamma=c^\gamma$


$S:=A_1\times \ldots\times A_n$


$\mathbb{Q}(\sqrt{-d})$


$A \subseteq B \subseteq C \subseteq A$


$\forall \epsilon >0,\quad\exists\delta:|x-a|<\delta\implies|f(x)-f(a)|<\epsilon$


$\lim\limits_{k \to \infty} \inf J(u_k)\geq J(v)$


$\overline{\{x\}}=\mathbb R$


$[t^n_i, t^n_{i + 1}]$


$(\overline{\mathbb{R}},\tau(\mathbb{\overline{R}})$


$F(x) = \int_{x-T}^x f(t)\,\mathrm{d}t$


$\frac{1-\gamma}{\gamma'-\gamma}\big[A-\frac{1-\gamma'}{1-\gamma}B\big]\in\mathscr{W}$


$w=\{\{a\},\{a,b\}\}$


$ q(T) = q((T^{*})^{*}) = [q(T^{*})]^{*} = \mathbf{0}^{*} = \mathbf{0}. $


$\int \frac{1}{x^2+x+1}\mathrm{d}x$


$\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$


$a-x,\:\:x\in \left(-\infty ,a\right)$


$E=F (a_{1},a_{2},a_{3},....,a_{n})$


$P[X< x |D_1] = 1$


$A_1\subset A_2, A_2\subset A_3, \ldots , A_{n-1}\subset A_n$


$\mathbb Z[i]\cong \mathbb Z[x]/(x^2+1).$


$ax+\sin x=L$


$[x,y] = {\bar x}^T H y.$


$P(m+1)\implies P(m)$


$n(\gamma,a)=-n(-\gamma,a).$


$A_0\times\dots\times A_n\times \Omega\times\Omega\times\dots$


$9^x - 6^x = 4^x.4^{1/2}$


$\begin{cases} x \equiv 1 \pmod4\\ x\equiv -1 \pmod{25} \end{cases}$


$ L = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(-x) = \lim_{x \to 0^+} -f(x) = -L. $


$(x-y, x+y)=(x,x)-(y,x)+(x,y)-(y,y)=(x,y)-(y,x)$


$ \gamma_1 \geq 0 \\ \gamma_2 \geq 0$


$\implies s_N=\frac{N(N+1)}{2}$


$(\mathbb{Z}[x] / (x^2-x))_U \cong \mathbb{Z}_2 \times \mathbb{Z}_2$


$P\{\sup_{t\geq 0} M_t > x \,| \,F_0\} = 1\wedge \frac{M_0}{x}$


$ \sigma_n:=\frac{x_1+\cdots+x_n}{n} $


$(1 + {x \over n})^n \geq 1 + x$


$G=\gamma_0(G) \geq \gamma_1(G) \geq \cdots \geq \gamma_{c}(G)=1,$


$-\dfrac{1}{2\pi}\int_{0}^{2\pi}\psi(s)\ln{\left(4\sin^2{\dfrac{t-s}{2}}\right)}ds\approx \displaystyle\sum_{i=0}^{2n-1}\psi(t_{j})R_{j}(t)$


$=\pi\int_0^\pi f(\sin t) dt- \int_0^\pi tf(\sin t) dt$


$\|y_n-x_n\|\leq r$


$|P(m) - P(m+t)| < \epsilon$


$\sqrt{\rho(A^t A)}$


$\gcd(a+b, a-b) = 1$


$\gamma _s(a)=\gamma (a),\quad \gamma _s(b)=\gamma (b)\quad \text{and}\quad \gamma _0=\gamma ,$


$\left|\int_{\gamma} f dz\right|\leqslant L(\gamma)|f|_\gamma.$


$dm=|ab|$


$v(r e^{i \theta}) = \frac{4r (1-r^2) \sin (\theta)}{(1-2r\cos (\theta)+r^2)^2}.$


$ |S(\gamma)(x,y)|^{2}\le S(\gamma)(x,x)S(\gamma)(y,y),\forall x,y\in \mathbb{R} $


$A(\sum x_n e_n) = \sum \lambda_n x_n e_n$


$ds=\sqrt {1+(\frac {dy}{dx})^2}dx$


$M = \frac{a+b+c+{\dots}n}{n}$


$ I_n = \int_0^\pi \frac{\sin(nt)}{\sin t} dt \\ I_{n+2}-I_n =\int_0^\pi \frac{\sin((n+2)t)-\sin(nt)}{\sin t} dt $


$\eta^+$


$\lim_{x\to c} g(x)=L=\lim_{x\to c} h(x)$


$\int_0^\pi \frac{x}{\sin{x}}dx=\int_0^{\frac{\pi}{2}} \frac{x}{\sin{x}}dx+\int_{\frac{\pi}{2}}^\pi \frac{x}{\sin{x}}dx$


$\mathbb{R^+}$


$A=\frac {1}{8} + \frac {1}{64} + ..... = \frac {1}{8} + \frac {1}{8} \cdot  (\frac {1}{8} + \frac {1}{64} + .....) = \frac {1}{8} + \frac {A}{8} $


$g(x) = \int_a^x f(t)\, dt$


$\mathbb{F}^*_{8} = (\mathbb{F}_2[x]/(x^3+x^2+1))^* $


$ \left[ \matrix{1 & 3/2 & -1/2 & 2 & | & a/2\cr                   0 & 1 & -1/7 & 12/7 & | & 3a/7 -2b/7\cr                   0 &  0   & 1 & -17/9 & | & -2a/3 + 4b/9 +7c/9 \cr                   0 &  0   &  0 &    0 & | & -a+b+c+d\cr}\right]$


$\Pr[X \ge x] \ge 1-k/q<)$


$ P(\{\omega \in \Omega: \lim_{n \to \infty}X_n(\omega) = X(\omega)\})=1. $


$ D : \mathbb{R}^{n+1} \to \mathbb{R}^n $


$ \left( {e^{\,x} } \right)^{\,{n \over m}}  = e^{\,\,{n \over m}\,x} \quad \left| \matrix{   \,n,m \in Z \hfill \cr    \;0 \ne m \hfill \cr}  \right. $


$\lim\limits_{n\rightarrow\infty}\frac{\sqrt{1+\frac{1}{n}}+\sqrt{1+\frac{2}{n}}+...+\sqrt{1+\frac{n}{n}}}{n}$


$ \forall s:\text{Re}(s)>0,\qquad \zeta(s)=\left(1-\frac{2}{2^s}\right)^{-1}\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^s}=\left(1-\frac{2}{2^s}\right)^{-1}\frac{1}{\Gamma(s)}\int_{0}^{+\infty}\frac{t^{s-1}}{e^t+1}\,dt $


$x=\frac{a+b}{a-b}$


$\int_{\gamma_{a,b} + \gamma_{b,z}} f = \int_{\gamma_{a,z}} f = F_a(z).$


$X=\begin{pmatrix}A&B\\C&D\end{pmatrix}$


$A_n = \frac{n(n+1)(n+2)}{6}$


$\frac{dx}{ds} = \frac{1}{\frac{ds}{dx}} = \frac{1}{\sqrt{1+ \big(\frac{dy}{dx}\big)^2}}$


$cov(X+Y,X-Y) = cov(X,X) + cov (X,-Y)+cov(Y,X)+cov(Y,-Y)$


$(n-1)C_n, (n-2)C_n, \ldots, \frac{n+1}{2}C_n$


$ e^x > \left(1 + \frac{x}{n}\right)^{n}. $


$\mathcal A= \mathcal A+x + (-x)$


$ S:= \{T_{1,1}, T_{2,1}, \ldots, T_{n,1}\} $


$\ell : ax + by = c$


$\int_{-\pi}^{\pi} \frac{\sin nx}{(1+2^x)\sin x}dx$


$f(x)=\frac{x}{\theta^2}\exp(\frac{-x}{\theta})$


$  T^\prime_{mix}(\varepsilon):=  \inf \Big  \{t \ge 0:   \max_{x,y \in \mathbb{Z}^d_n}\Big|\frac{1}{2}\Big(      \mathbb{P}(X_t = y | X_0=x)+\mathbb{P}(X_{t+1}= y | X_0=x)\Big)-       \pi(y)\Big|     \le \varepsilon \Big\} $


$a^b =alog(alog(\log(b)+\log(\log(a)))) $


$ |x^n - a^n| < |(a + \delta)^n - a^n| < |(a + a\delta)^n - a^n| = |a^n((1 + \delta)^n - 1)| $


$H \subset B \subset{\bar{B}} \subset G$


$ \begin{cases} x=-2 \\ x=-2 \\ y = -5 \end{cases}\Longleftrightarrow $


$ F_U(u) = \mathbb P(U\leq u) = \mathbb P \left[\left( 1+\frac{\gamma X}{\sigma} \right)^{-\frac{1}{\gamma}}\leq u\right] = F_{\gamma, \sigma}\left[ \frac{\sigma}{\gamma}\left( u^{-\gamma}-1 \right) \right] = 1-u. $


$R=A_1\times\cdots\times A_n$


$|f(z)| \le C ( 1+|z|^n)$


$y=\frac{(1+r)^n - 1}{(1+r)^n\times r}$


$ \begin{cases} x=t\\ y=t\\ z=0 \end{cases} $


$m,m',m''$


$\int_0^\infty f(x)=\infty$


$x \in V_0 \subseteq \overline{ V_0 } \subseteq U_0$


$ = \int_{-1}^{1}f(t)\delta(t-t_0)dt+\int_{-1}^{1}f(t)\delta(t+t_o)dt = f(t_0)+f(-t_0) = 0$


$\iff b-\dfrac{a+b}2>t-\dfrac{a+b}2\ge a-\dfrac{a+b}2\iff\dfrac{b-a}2>t-\dfrac{a+b}2\ge-\dfrac{b-a}2$


$Cov(I,X) = 0$


$6^{x} + 8^{x} + 9^{x} = 12^{x}$


$y =  log_a(b)$


$ \frac{\sqrt3 + 2}{2\sqrt3}(1+\sqrt3)^n + \frac{\sqrt3-2}{2\sqrt3}*(1-\sqrt3)^n $


$\infty > \sum_{k \in \mathbb{N}} \lVert  a_k-b_k \lVert \geq 1$


$f(a + b) = f(a) + f(b) + ab$


$f(x) =\frac{x}{9+x^2}$


$e(z,x,y) = (1,1,1)$


$P(k) \not \implies P(k+1)$


$x\sim y\iff xRy\land yRx$


$a x + b y + c x = d$


$\alpha+\sup_{\gamma<\beta}\{\gamma\}=\sup_{\gamma<\beta}\{\alpha+\gamma\}$


$\frac{x}{x-a} + x = \frac{b}{b-a}+ b$


$\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$


$ \frac{\pi}{\sin(\pi/p)} $


$ \frac{(n-p+1)!}{p!(n-2p+1)!} $


$\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{1}{n^k k^n}$


$\|x\| \leq \lim_{n\to \infty} \inf \|x_n\|$


$p_2 \mid p_1^d -1$


$as+bt=d$


$Cov(X,-Y)=-Cov(X,Y)$


$ f(n) = \binom{n}{k} => f(n) $


$\operatorname{Tr}(AABB)-\operatorname{Tr}(ABAB)\leq \operatorname{Tr}(A^{2})\left[ \operatorname{Tr}(ABB)-\operatorname{Tr}(AB)\operatorname{Tr}(AB)% \right]? $


$\lim_{x\downarrow \pi} f(x)=|\sin(\pi)+\cos(\pi)|=\lim_{x\uparrow \pi}f(x)=\lim_{x\to \pi}f(x)=1$


$(ax+by=c)$


$   A_1 \oplus \underbrace{\bigoplus_i (D_i' \oplus D_i'') \oplus \bigoplus_j D_j}_{C_1}   \simeq A_2 \oplus \underbrace{\bigoplus_j (D_j' \oplus D_j'') \oplus \bigoplus_i D_i}_{C_2} $


$|\sum x_n|\le\sum|x_n|$


$||A||_2 = \sqrt{r(A^TA)}$


$\sum_{n=1}^\infty na_n = \sum_{n=1}^\infty n(s_n-s_{n-1}) = \sum_{n=1}^\infty ns_n - \sum_{n=1}^\infty ns_{n-1} = \sum_{n=1}^\infty ns_n - \sum_{n=1}^\infty (n+1)s_n = -\sum_{n=1}^\infty s_n$


$ 1=1\cdot1=\left(a^{-1/2}a^{1/2}\right)\cdot\left(a^{1/2}a^{-1/2}\right) =a^{-1/2}\cdot\left(a^{1/2}a^{1/2}\right)\cdot a^{-1/2}=a^{-1/2}\cdot a\cdot a^{-1/2} $


$P(X_i = 1 | X_j = 1) \geq p^*$


$ \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} = \sum_{n\geq 1}\frac{H_n}{n 2^{n-1}}.$


$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s}=\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+...\Longleftrightarrow$


$\frac{2}{a} + \frac{4}{b} = 1$


$ P \left( X \geq n \right) = \begin{cases} \frac{1}{ {2}^{n - 2} } & n \geq 2 \\ 1 & n \leq 1 \end{cases} $


$\left\lfloor\frac n3\left\lfloor\frac{n-1}2\right\rfloor\right\rfloor-\varepsilon$


$cov(X,E(Y|X))=cov(X,Y)$


$(J_{(0,0)},J_{(1,0)},J_{(0,1)},\dots )$


$|X|^p \leq |X-X_n|^p + |X_n|^p$


$\frac{\zeta'(s)}{\zeta(s)}=log(2)+log(\pi)+\frac{\pi}{2}cot(\frac{\pi s}{2})-\frac{\Gamma'(1-s)}{\Gamma(1-s)}-\frac{\zeta'(1-s)}{\zeta(1-s)}$


$a = 2/4 = 3/6 = ... = (1*k)/(2*k)$


$f(x) = 1/27$


$ADD(x,y) = x + y$


$3^x+a^x\ge 6^x+9^x$


$\bigcup_rC_r =G\supset\ldots\supset C_r\supset C_{r+1}\supset \ldots C_0\supset C_1\supset C_2\supset\ldots \supset 0=\bigcap_rC_r\,.$


$ \forall \epsilon > 0, \exists \delta > 0, \forall x \in X : |x - y| < \delta \implies |f(x) - f(y)| < \epsilon $


$D = (1+x)^{n+1} - (1+x)^n - nx^2 $


$\binom{N}{2} = \Omega(N^2)$


$d(x,y)=\sqrt{\langle x,y \rangle}$


$a <x <b;c <y <d$


$\ell_p(t) = f(p) + f'(p)(t-p)$