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wu :: forums    riddles    putnam exam (pure math) (Moderators: SMQ, Eigenray, Icarus, towr, william wu, Grimbal)    Limit of ratio of rational sequences « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Limit of ratio of rational sequences  (Read 736 times) Michael Dagg Senior Riddler     Gender: Posts: 500 Limit of ratio of rational sequences   « on: Nov 7th, 2006, 5:35pm » Quote Modify Let    pn  and   qn  be sequences defined by setting p0 = 1,  p1  = 6,  q0 = 1 and   pn+1 = 6 p2n - 6 pn p2n-1 + 2 p4n-1,   qn+1 = pn+1 - p2n,  for n > 1.   Put  An = pn/qn, n > 0.   Show that lim n->oo An exists and find its value. IP Logged Regards, Michael Dagg Barukh Uberpuzzler     Gender: Posts: 2276 Re: Limit of ratio of rational sequences   « Reply #1 on: Nov 7th, 2006, 11:29pm » Quote Modify 1.2599...? IP Logged Michael Dagg Senior Riddler     Gender: Posts: 500 Re: Limit of ratio of rational sequences   « Reply #2 on: Nov 8th, 2006, 12:46pm » Quote Modify I knew you'd get it quickly.   Anyone else know the explicit form?   If so, just simply show by induction that     An  is less than _that_ number for all n > 0. « Last Edit: Nov 8th, 2006, 12:52pm by Michael Dagg » IP Logged Regards, Michael Dagg Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Limit of ratio of rational sequences   « Reply #3 on: Nov 8th, 2006, 7:14pm » Quote Modify hidden: Define tn = 1-1/An = (pn-qn)/pn = pn-12/pn. Let t=tn, t'=tn+1.  Then 1/t' = 6 - 6t + 2t2. Solving, A' = 1/(1-t')  = 1 + 1/(1/t'-1)  = 1 + 1/(5 - 6t + 2t2)  = 1 + A2/(5A2 - 6A(A-1) + 2(A-1)2)  = (2A2 + 2A + 2)/(A2 + 2A + 2). So An+1 = f(An), where f(x) = (2x2 + 2x + 2)/(x2 + 2x + 2)  = x + (2-x3)/(x2 + 2x + 2). From the above expression, it is clear that f(x) > x iff x3<2, and that f has a fixed point at x=21/3.  Since A0=1, we conclude that An = fn(1) converges to 21/3. IP Logged Michael Dagg Senior Riddler     Gender: Posts: 500 Re: Limit of ratio of rational sequences   « Reply #4 on: Nov 9th, 2006, 12:36pm » Quote Modify Hey, that's nice -- unexpected and simple too! IP Logged Regards, Michael Dagg Sameer Uberpuzzler Pie = pi * e     Gender: Posts: 1261 Re: Limit of ratio of rational sequences   « Reply #5 on: Nov 9th, 2006, 2:26pm » Quote Modify on Nov 9th, 2006, 12:36pm, Michael_Dagg wrote: Hey, that's nice -- unexpected and simple too!   Did you have some other method? IP Logged "Obvious" is the most dangerous word in mathematics. --Bell, Eric Temple Proof is an idol before which the mathematician tortures himself. Sir Arthur Eddington, quoted in Bridges to Infinity Michael Dagg Senior Riddler     Gender: Posts: 500 Re: Limit of ratio of rational sequences   « Reply #6 on: Nov 9th, 2006, 3:06pm » Quote Modify Sure.   edit: If you happen to have one of your own I'd to see it! « Last Edit: Nov 10th, 2006, 5:39pm by Michael Dagg » IP Logged Regards, Michael Dagg Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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