Search found 177 matches

by Riemann
Thu Feb 07, 2019 4:57 pm
Forum: General Mathematics
Topic: A sum!
Replies: 4
Views: 5797

Re: A sum!

First of all note that $10! = 2^8 \cdot 3^4 \cdot 5^2 \cdot 7$ hence $10!$ is not a perfect square, $10!$ has $(8+1)(4+1)(2+1)(1+1) = 270$ divisors. If $d \mid 10!$ , then there exists $p$ such that $dp =10!$ meaning that $p$ is also a divisor of $10!$. We also note that if one of $p, q$ is less tha...
by Riemann
Tue Jan 08, 2019 11:05 am
Forum: General Topology
Topic: A log metric
Replies: 0
Views: 6946

A log metric

Endow $(0, +\infty)$ with the following metric

$$d(x, y) = \left | \log \frac{x}{y} \right |$$

(i) Verify that $d$ is indeed a metric.

(ii) Is the sequence $a_n =\frac{1}{n}$ convergent under this metric? Give a brief explanation.

(iii) Examine if $(0, +\infty)$ is bounded under this metric.
by Riemann
Thu Dec 06, 2018 7:43 pm
Forum: Algebraic Structures
Topic: Find the number of homomorphism
Replies: 1
Views: 3519

Re: Find the number of homomorphism

Hi Ram_1729. Could you explain what $Q_8$ and $K_4$ stand for?
by Riemann
Thu Dec 06, 2018 7:43 pm
Forum: General Topology
Topic: $\mathbb{R}^2 \setminus \mathbb{Q} \times \mathbb{Q}$
Replies: 2
Views: 7370

Re: $\mathbb{R}^2 \setminus \mathbb{Q} \times \mathbb{Q}$

Thank you Ram_1729. Exactly! It was an exam's question!
by Riemann
Thu Nov 22, 2018 9:26 pm
Forum: General Mathematics
Topic: An inequality
Replies: 1
Views: 6308

An inequality

Let $x_1, x_2, \dots, x_n$ be $n \geq 2$ positive numbers other than $1$ such that $x_1^2+x_2^2+\cdots +x_n^2=n^3$. Prove that:

$$\frac{\log_{x_1}^4 x_2}{x_1+x_2}+ \frac{\log_{x_2}^4 x_3}{x_2+x_3}+ \cdots + \frac{\log_{x_n}^4 x_1}{x_n+x_1} \geq \frac{1}{2}$$
by Riemann
Tue Nov 06, 2018 4:48 pm
Forum: Algebraic Structures
Topic: Symetry group of Tetrahedron
Replies: 2
Views: 4706

Re: Symetry group of Tetrahedron

The tetrahedron is a regular solid with $4$ vertices and $4$ triangular faces. The symmetry group is the alternating group $\mathcal{A}_4$. The symmetry group of a cube is isomorphic to $\mathcal{S}_4$ , the permutation group on 4 elements. If we number the vertices of the cube from $1$ to $4$ and ...
by Riemann
Tue Oct 16, 2018 10:26 am
Forum: Linear Algebra
Topic: Computation of determinant
Replies: 0
Views: 4858

Computation of determinant

Let $A, B \in \mathcal{M}_{2 \times 2}$ be matrices with integer entries such that $AB = BA$ , $\det \left( A + B \right) =2$ and $\det \left( A^3 + B^3 \right) = 2^3$. Evaluate the determinant


$$\mathcal{D} = \det \left( A^2 + B^2 \right)$$
by Riemann
Sun Aug 12, 2018 8:27 pm
Forum: Calculus
Topic: Series with general harmonic number
Replies: 0
Views: 2988

Series with general harmonic number

Let $\mathcal{H}_n$ denote the $n$ - th harmonic number. It holds that $$\sum\limits_{n=1}^{\infty}\mathcal{H}_{pn}x^n = -\frac{1}{p}\sum\limits_{k=0}^{p-1} \frac{\ln \varphi_k}{\varphi_k}$$ where $p \in \mathbb{N}$ and $\displaystyle \varphi_k = \varphi_k(x) = 1 - \sqrt[p]{x}\exp\left(\frac{-2\pi i...
by Riemann
Wed Aug 01, 2018 9:40 am
Forum: Analysis
Topic: Multiplicity of root
Replies: 1
Views: 3590

Re: Multiplicity of root

Given the function $f(x)=e^x-x-1$ prove that $0$ is a zero of $f$ of multiplicity $2$. It suffices to prove that the limit $\displaystyle \lim \limits_{x \rightarrow 0} \frac{f(x)}{x^2}$ is finite. However, \begin{align*} \lim_{x\rightarrow 0} \frac{f(x)}{x^2} &= \lim_{x\rightarrow 0} \frac{e^x...
by Riemann
Wed Aug 01, 2018 9:35 am
Forum: General Mathematics
Topic: A sum!
Replies: 4
Views: 5797

Re: A sum!

Tolaso J Kos wrote:Evaluate the following sum:

$$\sum_{{\rm d}\mid 10!}\frac{1}{{\rm d}+\sqrt{10!}}$$
Source
Matha's notes.

$$\sum_{{\rm d}\mid 10!}\frac{1}{{\rm d}+\sqrt{10!}} = \frac{3\sqrt{7}}{112} $$

Full solution tomorrow morning !