My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 8th, 2017, 02:20 PM   #1
Senior Member
 
Joined: Dec 2006

Posts: 166
Thanks: 3

A limit has theoretical importance

If $a_1 = 1, a_{n+1}=\log(1+a_n),\;n\in\mathbb{N}^+$

Show that $\displaystyle{\lim_{n\to\infty}} \frac{a_n}{n a_n -2} = 0$

Last edited by greg1313; October 8th, 2017 at 05:19 PM.
elim is offline  
 
October 11th, 2017, 09:09 PM   #2
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 227
Thanks: 122

Math Focus: Dynamical systems, analytic function theory, numerics
Here are a few hints which should get you close enough to finish it on your own.

1. The sequence $a_n$ is monotone decreasing. To see this, Taylor expand log and factor to see that the tail is negative.
\[ a_{n+1} = \log(1+a_n) = a_n - \sum_{k = 2}^{\infty} \frac{1}{k!}a_n^k = a_n - \sum_{k = 1}^{\infty} a_n^{2k}(\frac{1}{(2k)!} + \frac{1} {(2k+1)!}a_n) < a_n \]

2. $a_n$ is non-negative and thus the sequence is bounded below so it must converge to a limit. To see this, look at the same Taylor series from 1.

3. Next prove that $a_n \to 0$. To see this, notice that the mean value inequality requires the estimate
\[ a_{n+1} = \log(1+a_n) \leq a_n \sup_{x \in (0,a_n)} \frac{1}{1+x}\]
to hold for all $n$. Since the sequence is Cauchy, it follows that
\[
\lim_{n \to \infty} \sup_{x \in (0,a_n)} \frac{1}{1+x} = 1
\]
which requires $a_n \to 0$.

The rest should be easy.
SDK is offline  
October 12th, 2017, 10:53 AM   #3
Senior Member
 
Joined: Dec 2006

Posts: 166
Thanks: 3

By Taylor, $0 = \log 1 < \log(1+x) = x-\frac{x^2}{2!(1+\theta x)^2}< x\;\small(x > 0 < \theta < 1)$
So $0< a_{n+1} < a_n\;(\forall n),\;\{a_n\}$ is decreasing and bounded below
hence convergent: $a_n\to A = \log(1+A)\ge 0\implies A=0$
i.e $a_n\searrow 0.$
By stolz and L'Hospital, ${\displaystyle\lim_{n\to\infty}} na_n={\displaystyle\lim_{n\to\infty}}\frac{n}{1/a_n}={\displaystyle\lim_{n\to\infty}}\frac{a_n\log (1+a_n)}{a_n-\log(1+a_n)}=2$
Let $\tau_n = \frac{na_n-2}{a_n} = n-\frac{2}{a_n}$, then $a_n=\frac{2}{n-\tau_n}\to 0,\;n-\tau_n\to\infty$
Now $\tau_{n+1}-\tau_n =1+\frac{2(\log(1+a_n)-a_n)}{a_n\log(1+a_n)}= \frac{1}{6}a_n + O(a_n^2)$. Thus for $\small 0< \lambda < \frac{1}{6}$,
$\tau_{n+1}-\tau_n > \lambda a_n\;(n\ge m)$ with some $m$. And so $\tau_{n+1}> \tau_m+{\displaystyle\sum_{k=m}^n}\large\frac{2 \lambda}{k-\tau_k}$
If $\{\tau_n\}_{n>m}$ were unbounded above, the RHS would be unbounded.
a contradiction. So $\tau_{n+m}\nearrow \infty.$ Consequently ${\displaystyle\lim_{n\to\infty}}\frac{a_n}{na_n -2}={\displaystyle\lim_{n\to\infty}}\frac{1}{\tau_n }=0$
elim is offline  
October 12th, 2017, 08:55 PM   #4
Senior Member
 
Joined: Dec 2006

Posts: 166
Thanks: 3

This is only a small step of $\displaystyle{\lim_{n\to\infty}\frac{n(n a_n -2)}{\log n}=?}\;\;(a_ > 0,\; a_{n+1}=\log(1+a_n)$

The challenge is: Numeric experiment is very difficult for this kind of sequence
elim is offline  
October 13th, 2017, 06:37 AM   #5
SDK
Senior Member
 
Joined: Sep 2016
From: USA

Posts: 227
Thanks: 122

Math Focus: Dynamical systems, analytic function theory, numerics
1. Your claim that $a_n \to 0$ is flawed. You can't assume because 0 is a lower bound that the sequence necessary converges to 0 e.g. $a_n = \frac{1}{2} + \frac{1}{n}$.

2. In fact, my sketch of the proof that $a_n \to 0$ above is incorrect so I would have to think a bit more whether or not this is even true. However, after some thought it seems likely this isn't true and the quotient goes to 0 due to the help from the factor of $n$ in the denominator.

3. L'hospital's doesn't apply to sequences and Stolz doesn't apply to Cauchy sequences. Your analysis is heading in the right direction but looking directly at a quotient is problematic if indeed $a_n \to 0$.

4. Why is this hard to numerically simulate? This would certainly help determine if $a_n \to 0$ or not.
SDK is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
importance, limit, theoratic, theoratical, theoretical



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Weight of Importance (scale) darev Number Theory 7 November 7th, 2013 01:15 PM
Win/loss ratio importance Algebra 1 February 15th, 2012 07:42 AM
importance of order in a sequence of numbers lorez Applied Math 5 September 26th, 2009 03:44 AM
Applying Monte Carlo method : importance sampling cristinelM Probability and Statistics 0 December 31st, 1969 04:00 PM





Copyright © 2017 My Math Forum. All rights reserved.