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We will first prove that \(a_n>0\) for all \(n\in\mathbb{N}\).

For \(n=1\), \(a_{1}=\alpha>0\), supposing that \(a_n>0\) for all \(n\in\mathbb{N}\), \(a_{n+1}=a_{n}{\alpha}^{a_{n}}>0\) which proves the induction.

If

\(\bullet\) \(\alpha=1\), then \(a_n=c\) for all \(n\in\mathbb{N}\) and some ...

Computation of \(\displaystyle \int_{0}^{\infty}\int_{0}^{\infty}\mathbb{e}^{-\frac{x^2+y^2}{2}}\sin(xy)\;\mathbb{d}x\;\mathbb{d}y\).

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\[\begin{eqnarray*}\int_{0}^{\infty} \int_{0 ...

A small comment, with a very brief examination we can easily deduce that $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\rm{sign}}(x)\,{\rm{sign}}(y)e^{-\frac{x^2+y^2}{2}}\sin(xy) \;dx\;dy=4\int_{0}^{\infty}\int_0^\infty e^{-\frac{x^2+y^2}{2}}\sin(xy)\;dx\;dy\,.$$

By definition \[\displaystyle \Gamma_{p}(x):=\frac{p^{x}p!}{x(x+1)(x+2)\cdot\ldots\cdot (x+p)}=\frac{p^{x}}{x\left(1+\frac{x}{1}\right)\left(1+\frac{x}{2}\right)\cdot\ldots\cdot\left(1+\frac{x}{p}\right)}\] and \[\Gamma(x):=\displaystyle\lim_{p\to \infty}\Gamma_{p}(x)\,.\]
\begin{align*}
p^{x}&=e ...

Prove that \(\displaystyle \lim_{n\to \infty} \mathcal{H}_{n}-\log(n)=-\int_{0}^{\infty}e^{-t}\log(t)\;dt\).

We have the series \(\displaystyle \sum_{k=1}^{\infty}\frac{(-1)^{k+1}x^{2k-1}}{(2k-1)(2k-1)!}\), now it is obvious that \(\displaystyle \limsup_{n\to\infty}\left|\frac{(-1)^{n+1}}{(2n-1)(2n-1)!}\right|=0\), so the radius of convergence is infinite and the interval we seek is \(\mathbb{R}\).

If ...

Prove that \(\sin\left(x^3\right)\) is a non-periodic function.

If \(z_1\) and \(z_2\) are the roots of the polynomial \(ax^2+bx+c\) with real coefficients and \(b^2<4ac\) prove that if \(z_1=\overline{z_{2}}\) then $$\arg\left(\frac{z_{1}}{z_{2}}\right)=2\arccos\left(\sqrt{\frac{b^2}{4ac}}\right)\,.$$

$$\begin{eqnarray*}f\left(\mathbb{E}[\mathbb{X}]\right) &=& f\left(\sum_{i}x_{i}\cdot p(x_i)\right)\\&=&f\left(\sum_{i}\left(\frac{1}{2}x_{i}+\left(1-\frac{1}{2}\right)x_i\right)\cdot p(x_i)\right) \\
&\leq& \frac{1}{2}\sum_{i} f(x_i)p(x_i) + \frac{1}{2}\sum_{i} f(x_i)\cdot p(x_i)\\ &=&\frac{1}{2 ...

An ice cream store has 20 different flavours. In how many ways can we order a dozen different ice cream cones, if each cone has 2 different flavours?