Search found 597 matches

by Tolaso J Kos
Wed Aug 30, 2017 8:18 am
Forum: Archives
Topic: A collection of problems in Analysis
Replies: 4
Views: 6833

Re: A collection of problems in Analysis

File updated to version $5$. As always your remarks are most welcome.
by Tolaso J Kos
Sun Aug 27, 2017 6:06 am
Forum: Number theory
Topic: Series with least common multiple.
Replies: 1
Views: 3513

Series with least common multiple.

Let ${\rm lcm}$ denote the least common multiple . Prove that for all $s>1$ the following holds:

$$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{{\rm lcm}^s(m, n)} = \frac{\zeta^3(s)}{\zeta(2s)}$$
by Tolaso J Kos
Wed Aug 16, 2017 12:09 pm
Forum: Calculus
Topic: A closed form of a hypergeometric series
Replies: 0
Views: 2616

A closed form of a hypergeometric series

The following result is new and is going to be published on Arxiv.org in the upcoming days with many more interesting results by Jacopo D' Aurizio who has actually proved it. Nevertheless , I am posting it here since it is interesting , challenging as well as approachable using only elementary tool...
by Tolaso J Kos
Wed Aug 09, 2017 2:48 pm
Forum: Real Analysis
Topic: A limit
Replies: 4
Views: 4649

Re: A limit

Unfortunately,

I do not remember where I had found this particular exercise and since I cannot recover the link this means I am unable to check for any particular typos that may have occured during typesetting.

Whoops!! Mea Culpa!
by Tolaso J Kos
Sun Aug 06, 2017 6:43 am
Forum: Real Analysis
Topic: Limit of an integral
Replies: 1
Views: 2479

Limit of an integral

The following exercise is just an alternative of IMC 2017/2/1 problem. It is quite easy but it's not a bad idea to have it here as well. Given the continuous function $f:[0, +\infty) \rightarrow \mathbb{R}$ such that $\lim \limits_{x \rightarrow +\infty} x^2 f(x) = 1$ prove that $$\lim_{n \rightarr...
by Tolaso J Kos
Wed Jul 26, 2017 10:50 pm
Forum: Calculus
Topic: A series involving Harmonic numbers
Replies: 2
Views: 4170

Re: A series involving Harmonic numbers

Let $\mathcal{H}_n$ denote the $n$-th harmonic number and consider the power series $$\sum_{n=1}^{\infty} \mathcal{H}_n \mathcal{H}_{n+1} x^n \quad , \quad -1 \leq x <1$$ Since $\mathcal{H}_{n+1} = \mathcal{H}_n + \frac{1}{n+1}$ then we have that \begin{align*} \sum_{n=1}^{\infty} \mathcal{H}_n \mat...
by Tolaso J Kos
Fri Jul 21, 2017 8:30 pm
Forum: Multivariate Calculus
Topic: Finite value
Replies: 0
Views: 2520

Finite value

Let $\mathcal{C} =[0, 1] \times [0, 1] \times \cdots \times[0, 1] \subseteq \mathbb{R}^n$ be the unit cube. Define the function $$f\left ( x_1, x_2, \dots, x_n \right )= \frac{x_1 x_2 \cdots x_n}{x_1^{a_1} + x_2^{a_2} + \cdots + x_n^{a_n}}$$ where $a_i$ arbitrary positive constants. For which values...
by Tolaso J Kos
Fri Jul 07, 2017 6:32 am
Forum: General Topology
Topic: On a Cauchy sequence
Replies: 1
Views: 6657

On a Cauchy sequence

Let $\mathbb{R}^+ =\{ x \in \mathbb{R}: x>0\}$. Endow it with the metric $${\rm d}(x, y) = \left| \frac{1}{x} - \frac{1}{y} \right|$$ Show that the sequence $a_n=n$ is a Cauchy one. Is the sequence $\frac{1}{n}$ a Cauchy one? Show that any sequence $a_n$ in $\mathbb{R}^+$ converges in $\mathbb{R}^+$...
by Tolaso J Kos
Sat Jun 24, 2017 9:58 am
Forum: Multivariate Calculus
Topic: Triple integral and ellipsoid
Replies: 1
Views: 2969

Triple integral and ellipsoid

Let ${\rm E}$ be the solid ellipsoid $${\rm E} = \left\{(x,y,z)\in\mathbb{R}^3 \; \bigg|\; \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1 \right \}$$ where $a > 0,\: b > 0,\: c > 0$ Evaluate $\displaystyle \iiint xyz \, {\rm d}(x, y, z)$ over: (a) the whole ellipsoid (b) that part of it ...
by Tolaso J Kos
Thu Jun 22, 2017 9:40 am
Forum: Linear Algebra
Topic: A symmetric matrix
Replies: 1
Views: 3714

Re: A symmetric matrix

Let $A$ be an $n \times n$ square matrix over a field $\mathbb{F}$ such that \begin{equation} A^2 =AA^{\top} \end{equation} Taking transposed matrices back at $(1)$ we get that \begin{align*} \left ( A^2 \right )^\top = \left ( A A^\top \right )^\top &\Rightarrow \left ( A^\top \right )^2 = \lef...