Search found 597 matches

by Tolaso J Kos
Fri May 18, 2018 12:03 pm
Forum: Archives
Topic: A collection of problems in Analysis
Replies: 4
Views: 6661

Re: A collection of problems in Analysis

A new version ( version 9 ) is out. Hope you find it entertaining. Awaiting to hear your feedback!
by Tolaso J Kos
Wed Feb 07, 2018 7:50 pm
Forum: Analysis
Topic: Analysis
Replies: 2
Views: 3834

Re: Analysis

Also worth noting Cantor's diagonal argument
by Tolaso J Kos
Wed Feb 07, 2018 7:49 pm
Forum: Analysis
Topic: Analysis
Replies: 2
Views: 3834

Re: Analysis

prove that $(0,1)$ is uncountable Assume that $(0,1)$ is countable. Then you can write $[0,1]=(x_n)_{n \geq 0}$. Do the following steps: - split $[0,1]$ into three equal parts $[0,1/3],[1/3,2/3],[2/3,1]$. Then $x_0$ is not in one of the given intervals. Denote it by $[a_0,b_0]$. - split $[a_0,b_0]$...
by Tolaso J Kos
Wed Feb 07, 2018 7:47 pm
Forum: Real Analysis
Topic: Real analysis
Replies: 3
Views: 5123

Re: Real analysis

Asis ghosh wrote:what can you say A if LUB A = GLB A ?
I beg your pardon?
by Tolaso J Kos
Tue Jan 02, 2018 2:20 pm
Forum: Real Analysis
Topic: Spivak, Michael : Calculus
Replies: 1
Views: 2992

Re: Spivak, Michael : Calculus

Yes Michael Spivak's book is a good one although it is densely written ...! I highly recommend it ...!
by Tolaso J Kos
Sun Oct 29, 2017 7:37 pm
Forum: Real Analysis
Topic: On an evaluation of an arctan limit
Replies: 1
Views: 3510

On an evaluation of an arctan limit

Evaluate the limit

$$\Omega = \lim_{n \rightarrow +\infty} \sum_{k=1}^{n} \frac{\frac{1}{n} \arctan \left ( \frac{k}{n} \right )}{1+2\sqrt{1+\frac{1}{n} \arctan \left ( \frac{k}{n} \right )}}$$

Dan Sitaru
by Tolaso J Kos
Wed Oct 25, 2017 8:02 pm
Forum: Real Analysis
Topic: Limit of a sequence
Replies: 0
Views: 2208

Limit of a sequence

Define the sequence $\{k_n\}_{n \in \mathbb{N}}$ recursively as follows

$$k_0 = \frac{1}{\sqrt{2}} \quad , \quad k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}$$

Evaluate the limit

$$\ell = \lim_{n \rightarrow + \infty} \left(\frac{4}{k_{n+1}}\right)^{2^{-n}}$$
by Tolaso J Kos
Mon Sep 25, 2017 10:22 pm
Forum: General Mathematics
Topic: On an operation
Replies: 0
Views: 2974

On an operation

Define

\begin{equation} x* y = \frac{\sqrt{x^2+3xy+y^2-2x-2y+4}}{xy+4} \end{equation}

Evaluate

$$\mathcal{V} = \left ( \left ( \cdots \left ( \left ( 2007 * 2006 \right )*2005 \right )* \cdots \right ) * 1 \right )$$
by Tolaso J Kos
Sat Sep 09, 2017 12:16 am
Forum: Meta
Topic: MathJaX Upgrade
Replies: 1
Views: 4042

MathJaX Upgrade

Greetings, today we upgraded to the latest stable release of MathJaX. This upgrade brings new features: A whole new set of macros thanks to mediawiki MathJaX extension . The revival of xypic.js which had been lost to the previous upgrade to 2.7.0 . Now we have once again the ability to draw commutat...
by Tolaso J Kos
Wed Aug 30, 2017 6:50 pm
Forum: Number theory
Topic: Is it an integer?
Replies: 0
Views: 2818

Is it an integer?

Investigating something on the serieses with Harmonic numbers the following occured. I do not have a solution , yet I would like to see one. Let $n \geq 1$ and denote as $\mathcal{H}_n$ the $n$ - th harmonic number. Let us also denote as ${\rm lcm} ( \cdot , \cdot )$ the least common multiple. Is i...