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by Riemann
Sat Dec 14, 2019 3:16 pm
Forum: Calculus
Topic: \(\sum_{n=2}^{\infty} \frac{(-1)^n}{n\log{n}}\)
Replies: 1
Views: 458

Re: \(\sum_{n=2}^{\infty} \frac{(-1)^n}{n\log{n}}\)

Basically it equals to

$$\int_1^\infty \left( 1+\left(2^{1-s}-1\right)\zeta(s) \right) \, \mathrm{d}s$$

However, the $\zeta$ function does not behave well under integrals. So, I would not expect a closed form to exist ... !
by Riemann
Wed Oct 23, 2019 7:58 pm
Forum: General Mathematics
Topic: An inequality
Replies: 1
Views: 764

Re: An inequality

The Engels form of the Cauchy – Schwartz inequality gives us: \begin{align*} \sum \frac{\log_{x_1}^4 x_2}{x_1+x_2} & \geq \frac{\left (\sum \log_{x_1}^2 x_2 \right )^2}{\sum (x_1+x_2)} \\ &= \frac{\left ( \sum \log_{x_1}^2 x_2 \right )^2}{2\sum x_1} \\ &\!\!\!\!\!\!\overset{\text{AM-GM}}{\geq } \fra...
by Riemann
Sun Oct 20, 2019 6:15 pm
Forum: General Mathematics
Topic: Arithmotheoretic limit
Replies: 0
Views: 545

Arithmotheoretic limit

Evaluate the limit:

$$\ell= \lim_{n \rightarrow +\infty} \frac{1}{n^2} \sum_{m=1}^{n} n \pmod m$$
by Riemann
Sat Oct 12, 2019 12:46 pm
Forum: Blog Discussion
Topic: A logarithmic Poisson integral
Replies: 1
Views: 680

Re: A logarithmic Poisson integral

It is closely related to another famous integral; namely $$\int_{0}^{\pi}\frac{\mathrm{d} \theta}{1-2a\cos \theta+a^2}\quad , \quad |a|<1$$ Evaluation of the integral: For $|a|<1$ we have successively: \begin{align*} \int_{0}^{\pi} \frac{{\rm d}x}{1-2a \cos x+a^2} &= \frac{1}{2} \int_{-\pi}^{\pi} \f...
by Riemann
Mon Sep 30, 2019 2:54 pm
Forum: Linear Algebra
Topic: Linear Projection
Replies: 2
Views: 417

Re: Linear Projection

Hi ,

I'm sorry but I do not understand what exactly you wrote down! Could you please elaborate?
by Riemann
Wed Sep 25, 2019 2:49 pm
Forum: General Mathematics
Topic: Inequality in a triangle
Replies: 0
Views: 176

Inequality in a triangle

Let $ABC$ be a triangle and denote $a, b, c$ the lengths of the sides $BC , CA$ and $AB$ respectively. If $abc \geq 1$ then prove that

$$\sqrt{\frac{\sin A}{a^3+b^6+c^6}} + \sqrt{\frac{\sin B}{b^3+c^6+a^6}} + \sqrt{\frac{\sin C}{c^3 + a^6+b^6}} \leq \sqrt[4]{\frac{27}{4}}$$
by Riemann
Sun Sep 22, 2019 7:51 pm
Forum: Meta
Topic: Welcome to the new and improved mathimatikoi.org
Replies: 7
Views: 1329

Re: Welcome to the new and improved mathimatikoi.org

Oh, also how about adding an Equation Editor button next to the Preview Button so that it links to an equation editor to speed up typesetting?
by Riemann
Sun Sep 22, 2019 5:36 pm
Forum: Linear Algebra
Topic: Linear Projection
Replies: 2
Views: 417

Linear Projection

Let $\mathcal{V}$ be a linear space over $\mathbb{R}$ such that $\dim_{\mathbb{R}} \mathcal{V} < \infty$ and $f:\mathcal{V} \rightarrow \mathcal{V}$ be a linear projection such that any non zero vector of $\mathcal{V}$ is an eigenvector of $f$. Prove that there exists $\lambda \in \mathbb{R}$ such t...
by Riemann
Sun Sep 22, 2019 5:31 pm
Forum: Meta
Topic: Welcome to the new and improved mathimatikoi.org
Replies: 7
Views: 1329

Re: Welcome to the new and improved mathimatikoi.org

I really like the add of the topic tags. Now every topic can be sorted into categories and be found easier. Hey , what about a live topic preview? That would be fantastic!
by Riemann
Sun Sep 22, 2019 5:29 pm
Forum: Complex Analysis
Topic: Contour integral
Replies: 0
Views: 209

Contour integral

Let $f$ be analytic in the disk $|z|<2$. Prove that: $$\frac{1}{2\pi i} \oint \limits_{\left | z \right |=1} \frac{\overline{f(z)}}{z-\alpha} \, \mathrm{d}z = \left\{\begin{matrix} \overline{f(0)} & , & \left | \alpha \right |<1 \\\\ \overline{f(0)} - \overline{f\left ( \frac{1}{\bar{\alpha}} \right...