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we are pleased to announce that the forum software has been upgraded to the latest version hardening the security of our website and resolving some issues noticed in previous releases.

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Greetings everyone,

we wanted to inform you that, unfortunately, the previous server hosting our forum unexpectedly went offline and ceased all operations without any prior notice. This sudden shutdown means that some of our recent data and posts may have been lost in the process.

The good news is ...

Evaluate the series

$$\mathcal{S} = \sum_{n=1}^{\infty} (-1)^{n-1} \ln \frac{n+1}{n}$$

Let $\mathbf{a} , \mathbf{b}, \mathbf{c}$ be three non coplanar vector. If $\displaystyle{\mathbf{a}' = \frac{\mathbf{b} \times \mathbf{c}}{\left [ \mathbf{a,b, c} \right ]} \; , \; \mathbf{b}' = \frac{\mathbf{c} \times \mathbf{a}}{\left [ \mathbf{a,b, c} \right ]} \; , \; \mathbf{c}' = \frac ...

Let $\mathcal{F}_n$ denote the $n$ -th Fibonacci number and $\mathcal{L}_n$ the $n$ – th Lucas. Prove that


$$\prod_{n=1}^{\infty} \left ( 1 + \frac{1}{\mathcal{F}_{2^n +1} \mathcal{L}_{2^n+1}} \right ) = \frac{3}{\varphi^2}$$

Greetings everyone,

we are pleased to announce that the forum software has been upgraded to the latest version hardening the security of our website. You will notice that many cosmetic things have been restored to normal. This new version is compatible with php 8.2 that our server is currently ...

Greetings,

we have updated the forum to its latest version phpbb 3.3.x. You will find that many bugs have been fixed in this latest version. We would like to also inform you that the ability to add tags has been restored and now it's working flawlessly. You can select among many different tags to ...

It follows from Taylor's theorem that $f(z)=\sum \limits_{n=0}^{\infty} c_n z^n$ and that the convergence is uniform. Thus,

\begin{align*} \frac{1}{2\pi i }\oint \limits_{|z|=1} \frac{\overline{f(z)}}{z-\alpha} \,\mathrm{d}z &=\frac{1}{2\pi i }\oint \limits_{|z|=1} \sum_{n=0}^{\infty} \frac ...

It holds that

$${\rm nul} (T_1 T_2) \leq {\rm nul} (T_1) + {\rm nul} (T_2)$$

where $T_1, \; T_2$ are the corresponding linear transformations.

Proof: The proof of the lemma is based on the rank - nullity theorem.


Based upon the above lemma we have that

\begin{align*}
{\rm rank} \left ( T_1 ...

The sum of $D(\sigma)$ over the even permutations minus the one over the odd permutations is the determinant of the matrix $A$ with entries $a_{i,j}=\vert i-j\vert$ and this determinant is known to be

$$\det A = (-1)^{n-1} (n-1) 2^{n-2}$$