1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Fibonacci Sequence converge exercise

  1. Sep 29, 2013 #1
    Let Fn denote the Fibonacci sequence.
    un is the sequence given by: un= Fn+1/Fn. Show that mod(un - [itex]\phi[/itex]) [itex]\leq[/itex][itex]\frac{1}{\phi}[/itex]mod(un-1-[itex]\phi[/itex]) and therefore mod(un - [itex]\phi[/itex]) [itex]\leq[/itex] [itex]\frac{1}{\phin-1}[/itex][/itex]mod(u1-[itex]\phi[/itex]) and then conclude un converges to [itex]\phi[/itex]

    I have tried with the identity [itex]\phi[/itex] = 1+ [itex]\frac{1}{\phi}[/itex] if anything came to light... And I tried dividing the mods but it got even more complicated.

    I can prove from the seocnd equation that un converges to [itex]\phi[/itex] as n-1 converges to infinity and thus 1/+inf =0, the right side becomes zero and we get mod(un - [itex]\phi[/itex]) [itex]\leq[/itex]0 which is the definiton of convergence. But I can't get from the first equation to the second. I don't know how to pass from un-1 to u1 and the part of the [itex]\phi[/itex]n-1. Can someone shed some light on this issue?
  2. jcsd
  3. Sep 29, 2013 #2


    User Avatar
    Homework Helper

    We have
    |u_n - \phi| \leq \frac{1}{\phi} |u_{n-1} - \phi|
    But since this is true for all n, we also have
    |u_n - \phi| \leq \frac{1}{\phi} |u_{n-1} - \phi| \leq \frac{1}{\phi} \frac{1}{\phi} |u_{n-2} - \phi| = \frac{1}{\phi^2} |u_{n-2} - \phi|
    and we can clearly keep going, picking up a factor of [itex]\phi^{-1}[/itex] each time, until we have a multiple of [itex]|u_1 - \phi|[/itex] on the right.
  4. Sep 29, 2013 #3
    Hmm, I didn't see that pattern. That is the same as making n-1=p and then doing the same for up and up-1.

    Nice, thank you :)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Fibonacci Sequence converge exercise
  1. Fibonacci sequence (Replies: 5)

  2. Fibonacci Sequence (Replies: 8)

  3. Fibonacci Sequence (Replies: 13)