Search found 597 matches
- Wed Aug 30, 2017 8:18 am
- Forum: Archives
- Topic: A collection of problems in Analysis
- Replies: 4
- Views: 9395
Re: A collection of problems in Analysis
File updated to version $5$. As always your remarks are most welcome.
- Sun Aug 27, 2017 6:06 am
- Forum: Number theory
- Topic: Series with least common multiple.
- Replies: 1
- Views: 5147
Series with least common multiple.
Let ${\rm lcm}$ denote the least common multiple . Prove that for all $s>1$ the following holds:
$$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{{\rm lcm}^s(m, n)} = \frac{\zeta^3(s)}{\zeta(2s)}$$
$$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{{\rm lcm}^s(m, n)} = \frac{\zeta^3(s)}{\zeta(2s)}$$
- Wed Aug 16, 2017 12:09 pm
- Forum: Calculus
- Topic: A closed form of a hypergeometric series
- Replies: 0
- Views: 3305
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...
- Wed Aug 09, 2017 2:48 pm
- Forum: Real Analysis
- Topic: A limit
- Replies: 4
- Views: 5489
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!
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!
- Sun Aug 06, 2017 6:43 am
- Forum: Real Analysis
- Topic: Limit of an integral
- Replies: 1
- Views: 2808
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 $$\lim_{n \rightarr...
- Wed Jul 26, 2017 10:50 pm
- Forum: Calculus
- Topic: A series involving Harmonic numbers
- Replies: 2
- Views: 4872
Re: A series involving Harmonic numbers
Let $\mathcal{H}_n$ denote the $n$-th harmonic number and consider the power series $$\sum_{n=1}^{\infty} \mathcal{H}_n \mathcal{H}_{n+1} x^n \quad , \quad -1 \leq x <1$$ Since $\mathcal{H}_{n+1} = \mathcal{H}_n + \frac{1}{n+1}$ then we have that \begin{align*} \sum_{n=1}^{\infty} \mathcal{H}_n \mat...
- Fri Jul 21, 2017 8:30 pm
- Forum: Multivariate Calculus
- Topic: Finite value
- Replies: 0
- Views: 3108
Finite value
Let $\mathcal{C} =[0, 1] \times [0, 1] \times \cdots \times[0, 1] \subseteq \mathbb{R}^n$ be the unit cube. Define the function $$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}}$$ where $a_i$ arbitrary positive constants. For which values...
- Fri Jul 07, 2017 6:32 am
- Forum: General Topology
- Topic: On a Cauchy sequence
- Replies: 1
- Views: 8378
On a Cauchy sequence
Let $\mathbb{R}^+ =\{ x \in \mathbb{R}: x>0\}$. Endow it with the metric $${\rm d}(x, y) = \left| \frac{1}{x} - \frac{1}{y} \right|$$ Show that the sequence $a_n=n$ is a Cauchy one. Is the sequence $\frac{1}{n}$ a Cauchy one? Show that any sequence $a_n$ in $\mathbb{R}^+$ converges in $\mathbb{R}^+$...
- Sat Jun 24, 2017 9:58 am
- Forum: Multivariate Calculus
- Topic: Triple integral and ellipsoid
- Replies: 1
- Views: 3411
Triple integral and ellipsoid
Let ${\rm E}$ be the solid ellipsoid $${\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 ...
- Thu Jun 22, 2017 9:40 am
- Forum: Linear Algebra
- Topic: A symmetric matrix
- Replies: 1
- Views: 4286
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...