
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
May 2nd, 2013, 10:18 AM  #1 
Newbie Joined: Jan 2013 Posts: 19 Thanks: 0  Proof of Convergence of a (recursive) sequence
Hi, I'd like to proof, if the following sequence converges; if yes, what's the limes? It's: a_1 > 0 und a_(n+1) = 1/(1+a_n) I thought I should show 1) The sequence is bounded (above or below) 2) The sequence is monotonous.. But I do not know where to begin =/ 
May 2nd, 2013, 06:40 PM  #2  
Senior Member Joined: Aug 2012 Posts: 2,324 Thanks: 715  Re: Proof of Convergence of a (recursive) sequence Quote:
The way you get started on a problem like this is to just plug in numbers to get a feel for the sequence. I took a = 2. Actually this was an accident, for some reason I thought it said a > 1 so I picked 2. But rereading, it says a > 0. So you should start by trying this for a = 1. a1 = 2 a2 = 1/(2+1) = 1/3 a3 = 1/(3/3 + 1/3) = 1/(4/3) = 3/4 a4 = 1/(4/4 + 3/4) = 1/(7/4) = 4/7 a5 = 1/(7/7 + 4/7) = 1/(11/7) = 7/11 ... The reason I wrote it out like that is that the process of adding 1 to a rational number is: 1 + (n/m) = (m/m) + (n/m) = (n+m)/m or "finding the common denominator" like they say in grade school. So you can begin to see the pattern. The numerators and denominators of your sequence are the terms of 3, 4, 7, 11, ..., but offset by one. I looked up 3, 4, 7, 11, 18, 29, 47, ... in OEIS, and it turns out to be the Lucas numbers (beginning at 2): L(n) = L(n1) + L(n2) http://oeis.org/A000032 Now I looked up Lucas numbers on Wikipedia. https://en.wikipedia.org/wiki/Lucas_number Lucas numbers are a special case of Lucas sequences. A Lucas sequence is a sequence in which a_(n+2) = a_n + a_(n+1). So it's actually a generalized Fibonacci sequence. In fact before reading the Wiki page on Lucas numbers, I perused the separate page on Lucas sequences ... https://en.wikipedia.org/wiki/Lucas_sequence In fact the concept of the Lucas sequence has as special cases the Fibonacci numbers and other important number sequences. (All these are integer sequences. Though I don't see why you wouldn't also be interested in letting the terms be real, or even complex. What you really care about is that each term is the sum of the previous two. In any event, now you put in 2, 1 as the first pair of numbers and you get 2, 1, 3, 4, 7, 11, 18, ... and these are the Lucas numbers. I'm going into all this detail because you asked how you get started on a problem like this. And the answer is that you can use traditional means, such as plugging in actual numbers and getting a feel for the situation and starting to see the pattern ... and you can combine that with modern methods such as looking things up on the Internet. Anyway it's pretty clear that for a = 2, your sequence is bounded by 0 and 1, and therefore converges. And I'm sure sure there must be some really cool closedform expression for the limit. Now you have to see if you can make sense of the general case with a > 0. I'm not saying my approach is the right one, it's probably not. But it's an example of how you can begin to get a handle on the problem.  

Tags 
convergence, proof, recursive, sequence 
Search tags for this page 
convergence of recursive sequences,recursion convergence,convergence of sequences proof,recursive sequence convergence,convergence of recursive sequence,complex series converge recursive,proof of alternating sequence converges to a number between a1 and a2,convergence recursive sequence,sequence recursive convergence proof,show that an alterna
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
recursive formula and convergence.  nappysnake  Calculus  6  November 30th, 2011 05:29 AM 
Solve Recursive Sequence  milagros  Applied Math  2  November 2nd, 2011 02:44 AM 
RECURSIVE SEQUENCE  milagros  Real Analysis  0  October 26th, 2011 11:46 AM 
Proof of recurrence sequence convergence  natkoza  Real Analysis  2  December 6th, 2010 01:20 PM 
Sequence convergence proof  zve5  Real Analysis  2  September 24th, 2008 02:32 PM 