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Sun Mar 12, 2023 3:03 pm
Forum: Calculus
Topic: An infinite product
Replies: 0
Views: 61

### An infinite product

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

$$\prod_{n=1}^{\infty} \left ( 1 + \frac{1}{\mathcal{F}_{2^n +1} \mathcal{L}_{2^n+1}} \right ) = \frac{3}{\varphi^2}$$
Sun Mar 12, 2023 2:52 pm
Forum: Meta
Replies: 1
Views: 475

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 runni...
Wed Nov 16, 2022 7:30 am
Forum: Meta
Replies: 1
Views: 475

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 ca...
Sun Apr 10, 2022 6:24 am
Forum: Complex Analysis
Topic: Contour integral
Replies: 1
Views: 3955

### Re: Contour integral

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{\ove...
Fri Nov 06, 2020 11:59 am
Forum: Linear Algebra
Topic: Rank of product of matrices
Replies: 1
Views: 4366

### Re: Rank of product of matrices

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 T_2 \rig...
Fri Nov 06, 2020 11:57 am
Forum: Linear Algebra
Topic: On permutation
Replies: 1
Views: 3167

### Re: On permutation

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

$$\det A = (-1)^{n-1} (n-1) 2^{n-2}$$
Fri Nov 06, 2020 6:50 am
Forum: Competitions
Topic: An equality with matrices
Replies: 1
Views: 3301

### Re: An equality with matrices

Let $A, B$ be elements of an arbitrary associative algebra with unit. Then: \begin{align*} \left ( A^{-1} +\left ( B^{-1} - A \right )^{-1} \right )^{-1} &= \left ( A^{-1} \left ( B^{-1} - A \right )\left ( B^{-1} - A \right )^{-1} + A^{-1} A \left ( B^{-1} - A \right )^{-1} \right )^{-1} \\ &am...
Fri Nov 06, 2020 6:36 am
Forum: Algebraic Structures
Topic: Sum equals to zero
Replies: 1
Views: 2242

### Re: Sum equals to zero

Let us suppose that $|\mathcal{G}| = \kappa$ and $x= \frac{1}{\kappa} \sum \limits_{g \in \mathcal{G}} g$. We note that for every $h \in \mathcal{G}$ the depiction $\varphi: \mathcal{G} \rightarrow \mathcal{G}$ such that $\varphi(g)=h g$ is $1-1$ and onto. Thus: \begin{align*} x^2 &=\left ( \f...
Fri Nov 06, 2020 5:28 am
Forum: Algebraic Structures
Topic: Isomorphic groups
Replies: 1
Views: 2577