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September 24th, 2008, 11:25 AM  #1 
Newbie Joined: Sep 2008 Posts: 3 Thanks: 0  Sequence convergence proof
So i have this problem that asks to prove that the following sequence convergences, and find it its limit. a1=sqrt (1) a2= sqrt (1+sqrt1) a3= sqrt ( 1+sqrt(1+sqrt(1))) a4= sqrt (1+sqrt(1+sqrt(1+sqrt(1)))) 
September 24th, 2008, 12:23 PM  #2 
Member Joined: Sep 2008 Posts: 46 Thanks: 0  Re: Sequence convergence proof
First. You show that this sequence is strictly increasing (use induction). If sqrt 1 < sqrt (1+sqrt 1) means s1<s2, then hypothesize sn < sn+1. Use induction again to show that it is bounded above. ex. s1 = sqrt 1 < 2. If sn <2 then sn+1 = sqrt(1+sn) < sqrt (1+2) < 2... This isn't the tightest form of this, but we've shown that {sn} is increasing and bounded above. This means that {sn} must converge to some number s. We know that it is less than or equal to whatever you show it to be bounded by. Be careful about when it could be equal to this upper bound. Note we could have said that it is bounded above several numbers. This doesn't mean the sequence will converge to it. 
September 24th, 2008, 02:32 PM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Sequence convergence proof
The above post shows how to prove that the sequence has a limit. To show what the limit is, substitute the expression into itself: x = sqrt(1 + sqrt(1 + sqrt(1 + ...))) x = sqrt(1 + x) 

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