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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
 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) Tags convergence, proof, sequence ,
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# convergence proof

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