I'm trying to make this look nicer and more readable (brackets tall enough, fraction large enough to easily read, etc).  $((\frac{(x(1-\delta_{1}))}{(z(1-\delta_{3}))}(1-\delta_{4}))(y(1-\delta_{2})))(1-\delta_{5})$
101373
If I have a document like this:  \documentclass[landscape, 12pt]{report}  \usepackage[landscape]{geometry}  \usepackage[utf8x]{inputenc} \usepackage[T1]{fontenc}  \usepackage{amsmath, amssymb, graphics, setspace} \newcommand{\mathsym}[1]{{}} \newcommand{\unicode}[1]{{}} \newcounter{mathematicapage}  \begin{document}    \begin{equation}\label{Equation:Naive_Bayes_Classifier}     P\left(H_h|E_1,E_2,\ldots ,E_e,\ldots    E_{\mathbb{E}}\right)=\frac{P\left(H_h\right) P\left(E_1|H_h\right)    P\left(E_2|H_h,E_1\right) \text{$\ldots $P}     \left(E_e|H_h,E_1,E_2,\ldots ,E_{e-1},E_{e+1} \ldots, E_{\mathbb{E}}\right) \text{$\ldots    $P}     \left(E_{\mathbb{E}}|H_h,E_1,E_2,\ldots ,E_{\mathbb{E}-1}\right)}{ P\left(E_1,E_2,\ldots,E_e, \ldots    ,E_{\mathbb{E}}\right)} \end{equation} \end{document}   How can I auto adjust this equation to appear in the entire page, using all horizontal space? (Without badboxes or smaller than the width of the page?)  If is there a kind of scale to use in equations, giving explicity how greater the equation should become, this solver my problem too.
109778
$$\begin{aligned}f_Y(y_i)&amp;=\frac{{\theta_i}^2\left(y_i+\theta_i+2\right)}{\left(1+\theta_i\right)^{y_i+3}}\\ &amp;=\exp\ \log\left[\frac{{\theta_i}^2\left(y_i+\theta_i+2\right)}{\left(1+\theta_i\right)^{y_i+3}}\right]\\ &amp;=\exp\left[\log\left(\frac{{\theta_i}^2\left(y_i+\theta_i+2\right)}{\left(1+\theta_i\right)^{y_i+3}}\right)\right]\\ &amp;=\exp\left[\log\left({\theta_i}^2\right)+\log\left(y_i+\theta_i+2\right)-\log\left(1+\theta_i\right)^{y_i+3}\right] \end{aligned}$$ Need to write in the form $$f(y) = \exp{ [y(\theta) - b(\theta)]/ a(thi) + c(y, thi)}$$ I am actually stuck here and I am not sure about whether it can be written as canonical form or not. I have read somewhere that mixed distributions are not members of the exponential family.
31000
$$x + \frac1x + \cfrac1{x + \frac1x} + \cfrac1{x + \frac1x + \cfrac1{x + \frac1x}} +\cfrac1{x + \frac1x + \cfrac1{x + \frac1x} + \cfrac1{x + \frac1x + \cfrac1{x + \frac1x}} } + \cdots $$  How can this series be simplified? Consider me as a high school graduate.
69759
I'm a PreCalculus student trying to find a rigorous proof that $\displaystyle\frac{1}{3} = 0.333\ldots$, but I couldn't find it. I think (just think) that this proof would start by proving that  $\displaystyle\sum_{i=1}^{\infty}3\cdot10^{-i} = \frac{1}{3}$. My guesses (assuming that proving that $\displaystyle\sum_{i=1}^{\infty}\left(\frac{1}{10}\right)^i$ converges is trivial):  $\displaystyle\sum_{i=1}^{\infty}3\cdot10^{-i} = 3\cdot\sum_{i = 1}^{\infty}10^{-i} = 3\cdot\sum_{i=1}^{\infty}\left(\frac{1}{10}\right)^i = 3\cdot\left(\frac{1}{1 - \frac{1}{10}}-1\right) = 3\cdot\left(\frac{10}{9}-1\right) = \frac{1}{3}$.  Questions: is this completely rigorous? Which flaws could be found in this proof? How can I improve it?  PS. I'm not sure how to tag this. Feel free to edit, if necessary.
43311
If $X \sim \text{Gamma}(\alpha,\beta)$, how would I go about finding $E\left(\frac 1{X^2}\right)$?
12873
This is the question I'm stumbling with:   When $|\alpha| &lt; 1$ and $|\beta| &lt; 1$, show that:  $$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| &lt; 1$$    The chapter that contains this question contains (among others) the triangle inequalities:   $$\left||z_1| - |z_2|\right| \le |z_1 + z_2| \le |z_1| + |z_2| $$  I've tried to use the triangle inequalities to increase the dividend and/or decrease the divisor:  $$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| &lt; \cfrac{|\alpha| +|\beta|}{\left|1-|\bar{\alpha}\beta|\right|}$$  But it's not clear if or why that would be smaller than one.  I've also tried to multiply the equation by the conjugated divisor $\cfrac{1-\alpha\bar{\beta}}{1-\alpha\bar{\beta}}$, which gives a real divisor, but the equation does not appear solvable.  Any hint would be much appreciated.
97347
Here is the code which has an equation in the form of \[... \]:  \documentclass[12pt]{report} \usepackage{braket} \usepackage{amssymb} \usepackage{amsmath} \begin{document} \textbf{Proof-} Every element of $\Delta(G)\Delta(G,N)\subset N'$ can be written as \((x-1)y(n-1)\) where \(x,y\in G\) and \(n\in N\). Now, because we can write \((x-1)y=xy-y=(xy-1)-(y-1)\) we also can write every $(g-1)$ in the form : \[g-1=\sum_{i,j}a_{ij}(x_i-1)(n_j-1), \hspace{2pt}\ a_{ij}\in \mathbb{Z}, x_i\in G, n_j\in N \] \end{document}   My question is how to label the last formula as (*)? All the answers I can find has aligned equations, but in this case I only have one \[...\] equation which I want to label. One way is to use \hspace and then put label,  but is there any other way?
39231
$$ \arctan\left(\frac{1}{1+n+n^2}\right)$$  My professor wrote this as  $$\arctan(n+1) - \arctan(n)$$   I don't understand how this expression is right?
28412
$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$   I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
110685
Evaluate : $$\int_{0}^{+\infty }{\left( \frac{x}{{{\text{e}}^{x}}-{{\text{e}}^{-x}}}-\frac{1}{2} \right)\frac{1}{{{x}^{2}}}\text{d}x}$$
59359
Does $\displaystyle\lim_{n\to \infty}\ \frac{1+\sqrt[2]{2}+\sqrt[3]{3}+\cdots+\sqrt[n]{n}}{n}=1$?   Thanks a lot for your time and help.
96139
Could you please help me computing    $\Big\lfloor\underbrace{\sqrt{2017+\sqrt{2017+...+\sqrt{2017}}}}_\text{2017 roots}\Big\rfloor$?
121843
$$\frac{dx(t)}{dy(t)}=\frac{\alpha x(t) - \beta x(t) y(t)}{-\gamma y(t) + \delta x(t)y(t)}$$  How would one simplify this fraction? Maybe the chain rule could be of any use, but I don't see how.
103696
Could someone give me a hint? I would like to continue to attempt it. The limit to evaluate that I would like a hint on is:  $$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$$
34259
I think I have found the final answer $\left(4 \left[\frac{2t \cosh(4t) - \sinh(4t)}{t^2}\right]\right)$ but need to verify it.
33077
$\displaystyle \int \left( \frac{1}{x^2+3} \right)\; dx$  I've let $u=x^2+3$ but can't seem to get the right answer.  Really not sure what to do.
104001
Writing $\left(\mathbf Z\big/2\oplus\mathbf Z\big/4\oplus\mathbf Z\big/8\right)\oplus\left(\mathbf Z\big/9\oplus\mathbf Z\big/27\right)$ in a different way   If $M$ is equal to above how can I find the ideals,(of course of the form $m\mathbf Z$) $I_1,\dots I_n$ such that $I_1\supset I_2\supset \dots I_n$ and $M\simeq\mathbf Z\big/I_1\oplus\dots\oplus\mathbf Z\big /I_n$  (there should be only $1$ way of selecting such ideals, but I don't know how)