My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Thanks Tree2Thanks
  • 1 Post By Extrazlove
  • 1 Post By Extrazlove
Reply
 
LinkBack Thread Tools Display Modes
February 13th, 2019, 02:07 PM   #1
Banned Camp
 
Joined: Feb 2019
From: Casablanca

Posts: 23
Thanks: 2

series vs theorem

Hello

My series is

$R_3=2$
$\displaystyle R_{n+1}=\frac{R_n}{\cos\left(\frac{\pi}{n}\right)}$

I will demonstrate why it is divergent.

To calculate the limit from n to infinity we must first define the n,
so I'm going to calculate it

Can find from the formula that n = pi / arccos (Rn / (Rn + 1)) so Rn / (Rn + 1) is different from 1 to define the n.
The sequence is increasing, so we have (Rn + 1) / Rn > 1 so the result is divergent according to the d'Alembert criterion.


Can demonstrate that this sequence is convergent with the first comparison theorem.
Convergence can be easily demonstrated, but with tools that go beyond high school.

We have for $ n \geq 3 $: $$ R_n = \frac2 {\displaystyle\prod_{k = 3}^{n-1} \cos \left (\frac {\pi} {k} \right)} $$ and the series of logarithms $\displaystyle \sum_{k \geq3} - \ln \left (\cos \left (\frac {\pi} {k} \right) \right) $ converges since its general term is equivalent to $\displaystyle \frac {\pi ^ 2} {2k^2} $. The infinite product converges to a non-zero limit.

Last edited by skipjack; February 14th, 2019 at 06:42 AM. Reason: to improve markup
Extrazlove is offline  
 
February 13th, 2019, 03:19 PM   #2
Senior Member
 
Joined: Dec 2015
From: iPhone

Posts: 474
Thanks: 73

Yes but the math codes are not showing well .
idontknow is offline  
February 13th, 2019, 03:24 PM   #3
Banned Camp
 
Joined: Feb 2019
From: Casablanca

Posts: 23
Thanks: 2

see message 6

Une suite convergente ? / Enigmes, casse-têtes, curiosités et autres bizarreries / Forum de mathématiques - Bibm@th.net
Thanks from idontknow
Extrazlove is offline  
February 13th, 2019, 03:51 PM   #4
Banned Camp
 
Joined: Feb 2019
From: Casablanca

Posts: 23
Thanks: 2

Then the sequence is convergent or divergent.
correctly define it in which it is divisive and it contradicts a theorem.
Extrazlove is offline  
February 14th, 2019, 01:25 AM   #5
Banned Camp
 
Joined: Feb 2019
From: Casablanca

Posts: 23
Thanks: 2

look at this discussion to understand(us google traduction).
In mathematics or computer science, you must define the variables before using them (n Rn x ....)
In the 2 demonstration he plays with an indeterminate form of n so that demonstration is false.
Une suite convergente ? / Enigmes, casse-têtes, curiosités et autres bizarreries / Forum de mathématiques - Bibm@th.net
Thanks from idontknow
Extrazlove is offline  
February 14th, 2019, 05:35 AM   #6
Senior Member
 
Joined: Dec 2015
From: iPhone

Posts: 474
Thanks: 73

What does it mean and how to get there or prove it ?
$\displaystyle \sum_{k\geq 3} -\ln(\cos (\frac{\pi}{k}))\; $ converges since the general term is equivalent to $\displaystyle \frac{\pi^{2}}{2k^2 }$ .
idontknow is offline  
February 14th, 2019, 01:54 PM   #7
Banned Camp
 
Joined: Feb 2019
From: Casablanca

Posts: 23
Thanks: 2

Here is a bizarre series I will show that it diverges and normally the comparison theorem shows who it converges is equivalent to $\displaystyle \frac{\pi^{2}}{2k^2 }$.
R3 = 2

Rn + 1 = rn / cos (pi / n)
I will demonstrate who she is divergent

To calculate the limit of n to infinity, we must first define the n for not falling on things that do not exist like 1/0.

So I'm going to calculate it

In Can find from the formula that n = PI / arcos (Rn / Rn + 1) So Rn / Rn + 1 is different from 1 to define well the n and not to fall on an absurdity 1/0 which leads to false calculations .

The sequence is thus increasing Rn + 1 / Rn> 1 so the result is divergent according to the Dalembert criterion so as not to fall on a 1/0 absurdity.


Who is the valid demonstration?
Extrazlove is offline  
February 14th, 2019, 01:57 PM   #8
Banned Camp
 
Joined: Feb 2019
From: Casablanca

Posts: 23
Thanks: 2

Quote:
Originally Posted by idontknow View Post
What does it mean and how to get there or prove it ?
$\displaystyle \sum_{k\geq 3} -\ln(\cos (\frac{\pi}{k}))\; $ converges since the general term is equivalent to $\displaystyle \frac{\pi^{2}}{2k^2 }$ .

It is a famous mathematician who has dismantled it here
Une suite convergente ? / Enigmes, casse-têtes, curiosités et autres bizarreries / Forum de mathématiques - Bibm@th.net
Extrazlove is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
series, theorem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Riemann series theorem and integrals littlelisper Real Analysis 2 December 1st, 2013 10:14 PM
Series Divergence Theorem aaron-math Calculus 5 December 17th, 2011 06:27 PM
Sum of series and residue theorem SonicYouth Complex Analysis 8 April 20th, 2011 08:41 AM
Deriving the series of e using binomial theorem laryngex Calculus 4 June 6th, 2010 11:25 PM
Liouville Theorem&Taylor Series WannaBe Complex Analysis 0 January 16th, 2010 01:16 AM





Copyright © 2019 My Math Forum. All rights reserved.