Search found 597 matches

by Tolaso J Kos
Tue Nov 10, 2015 4:05 pm
Forum: Calculus
Topic: Logarithmic integral
Replies: 2
Views: 3071

Re: Logarithmic integral

Hey Grigoris, thank you for your solution. Here is another one which actually is able to crack with more ease the more general case: $$\int_0^1 \frac{\ln (1-x) \ln^k x}{1-x}\, {\rm d}x, \;\;\; k \geq 1$$ We have successively: $$\begin{align*} \int_{0}^{1}\frac{\ln (1-x)\ln x}{1-x}\, {\rm d}x &=-...
by Tolaso J Kos
Tue Nov 10, 2015 4:02 pm
Forum: Number theory
Topic: Irrational number
Replies: 1
Views: 5885

Irrational number

Let $N \in \mathbb{N} \mid N>1$. Prove that the number:

$$\mathcal{N}= \sqrt{1\cdot 2 \cdot 3 \cdot 4 \cdots (N-1)\cdot N}$$

is irrational.
Hidden message
May we have an alternate proof without Bertrand's postulate,or is this impossible?
by Tolaso J Kos
Tue Nov 10, 2015 3:59 pm
Forum: Number theory
Topic: Five last digits of number
Replies: 4
Views: 4452

Five last digits of number

Find the five last digits of the number:

$$\mathscr{N}=\underbrace{9^{9^{9^{\cdot^{\cdot^{\cdot^{\cdot^{9}}}}} }}}_{1001 \;\; \rm {nines}}$$
by Tolaso J Kos
Tue Nov 10, 2015 2:09 pm
Forum: General Mathematics
Topic: There do not exist points
Replies: 2
Views: 2910

There do not exist points

Prove that there do not exist four points in $\mathbb{R}^2$ whose pairwise distances are all odd integers.
by Tolaso J Kos
Tue Nov 10, 2015 2:03 pm
Forum: Calculus
Topic: Logarithmic integral
Replies: 2
Views: 3071

Logarithmic integral

Evaluate the integral:

$$\int_0^1 \frac{\ln (1-x) \ln x}{1-x}\, {\rm d}x$$
by Tolaso J Kos
Mon Nov 09, 2015 5:18 pm
Forum: General Mathematics
Topic: A sum!
Replies: 4
Views: 5642

A sum!

Evaluate the following sum:

$$\sum_{{\rm d}\mid 10!}\frac{1}{{\rm d}+\sqrt{10!}}$$
Source
Matha's notes.
by Tolaso J Kos
Mon Nov 09, 2015 5:16 pm
Forum: Real Analysis
Topic: Convergence of a series
Replies: 1
Views: 2674

Convergence of a series

Let \( a_n =\underbrace{\sin \left ( \sin \left ( \sin \cdots (\sin x) \cdots \right ) \right )}_{n \; \rm {times}} \) and \( x \in (0, \pi/2) \). Examine if the series: $$ \mathcal{S}=\sum_{n=1}^{\infty} a_n $$ converges. Do the same question for the series: \( \displaystyle \mathcal{S}=\sum_{n=1}^...
by Tolaso J Kos
Mon Nov 09, 2015 5:08 pm
Forum: Calculus
Topic: Double integral
Replies: 1
Views: 2552

Double integral

Let $\displaystyle D= \left\{ \dfrac{1}{2} \leq x^2+y^2\leq 1, \; x^2+y^2-2x \leq 0 \;\;\; y \geq 0 \right\}$. Evaluate the double integral:

$$\iint \limits_{D} \frac{\log (x^2+y^2)}{\sqrt{x^2+y^2}}\; {\rm d} x\; {\rm d} y$$
Hidden Message
I don't have a solution.
by Tolaso J Kos
Mon Nov 09, 2015 5:04 pm
Forum: General Mathematics
Topic: Finite sums
Replies: 0
Views: 1888

Finite sums

These are two well known sums, but let them exist here as well.

a) $\displaystyle \sum_{k=1}^{m} \tan^2\left(\frac{k\pi}{2m+1}\right) = m(2m+1)$

b) $\displaystyle \sum_{k=1}^{n-1}\tan^{2}\frac{k \pi}{2n} = \frac{(n-1)(2n-1)}{3}$
by Tolaso J Kos
Mon Nov 09, 2015 2:57 pm
Forum: General Topology
Topic: Rendezvous value
Replies: 3
Views: 3527

Re: Rendezvous value

Hi Demetres, how about if we replace complete with compact? I think that the exercise now would be correct. I was thinking about this in the last $3$ days and I came up to the same conclusion that this cannot hold in $\mathbb{R}$ as you said. But I think I have a huntch that replacing complete with ...