It is currently Tue Feb 20, 2018 10:27 am


All times are UTC [ DST ]




Post new topic Reply to topic  [ 2 posts ] 
Author Message
 Post subject: Convergence of a series
PostPosted: Mon Nov 09, 2015 5:16 pm 
Administrator
Administrator
User avatar

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 827
Location: Larisa
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: \( \displaystyle \mathcal{S}=\sum_{n=1}^{\infty}a_n^r , \;\; r\in \mathbb{R}^+ \).

_________________
Imagination is much more important than knowledge.
Image


Top
Offline Profile  
Reply with quote  

PostPosted: Wed Jul 26, 2017 4:58 am 

Joined: Tue Nov 24, 2015 7:47 pm
Posts: 13
First $ a_{n+1}=sin(a_n) \leq a_{n} $
Since $ a_1 \in (0, \frac{\pi}{2} ) $ then using induction one can prove $ a_n \in (0, \frac{\pi}{2}) $

Hence $ a_n $ being bounded from below $ \lim_{n \to \infty} a_n=0 $
To estimate what the sequence feels like we compute the following limit

$$ \lim_{n \to \infty} n a_n^2 $$

But to use cezaro stolz lemma i will compute this one instead :
$ \lim_{n \to \infty} \frac{1}{n a_n^2}=\lim_{n \to \infty} \frac{a_n^2-a_{n+1}^2}{a_{n+1}^2a_n} $

So in order to compute that we compute it's "continuous" version and then we jump on the discrete case using continuity .

$ \lim_{x \to 0} \frac{x-sin^2x}{xsin^2(x)} $ dividing both sides with $ x^3 $ we and after using DLH rule several times we get that $ \lim_{x \to 0} \frac{x-sin^2x}{xsin^2(x)}=\frac{1}{3} $

Hence $ \lim_{n \to \infty}\frac{1}{n a_n^2} =\frac{1}{3} \Rightarrow \lim_{n \to \infty} n a_n^2 = 3 \Rightarrow \lim_{n \to \infty} \sqrt{n} a_n=\sqrt3 $

So using comparison test with $ \frac{1}{\sqrt{n}} $ our series diverges .

For the second part comparing with $ (\frac{1}{\sqrt{n}})^r $ we get that it converges if and only if $ 2r>4 \Leftrightarrow r>2 $ and diverges if $ r \leq 2 $


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 2 posts ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net