1. Not finding help here? Sign up for a free 30min 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!

Help to find limit

  1. Nov 8, 2007 #1
    Please, help to find limit:
    [tex]\lim_{n \rightarrow \infty} na(n)[/tex], where
    Thanks for any ideas!
  2. jcsd
  3. Nov 8, 2007 #2


    User Avatar
    Science Advisor

    If the limit exists, then a(n) is going to zero at large n, which can be used to simplify the right-hand side.

    Also, at large n, a finite-difference equation can be well approximated by a differential equation.
  4. Nov 8, 2007 #3
    Thanks. With the help of your post, Avodyne, I found that limit equals 1. Is it correct?
    Sorry, but I'm not sure.
    Last edited: Nov 8, 2007
  5. Nov 8, 2007 #4


    User Avatar
    Science Advisor

    Yes, this is correct!
  6. Nov 8, 2007 #5
    But actually, I didn't use differential equation :)
    It's easy to show that [tex]a_n[/tex] is going to zero at large n, but remains postive. So:
    [tex]\frac{1}{a_{n+1}}=\frac{1+a_n-\frac{1}{6}a_n^3+o(a_n^3)}{a_n} =\frac{1}{a_n}+1+b_n[/tex], where [tex]b_n \rightarrow 0[/tex], when [tex]n \rightarrow \infty[/tex].
    Use this we can obtain:
    When n is going to infinity, we have: (using well-known [tex]\lim_{n \rightarrow \infty}\frac{b_1+b_2+...+b_n}{n}=0[/tex], where each of b is going to zero with large n)
    [tex]\lim_{n \rightarrow \infty} \frac{1}{na_{n+1}}=0+1+0=1[/tex]

    In any case, thank you very much, Avodyne. :)
  7. Nov 8, 2007 #6


    User Avatar
    Science Advisor

    Very nice. Your analysis is more rigorous than mine was.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Help to find limit
  1. Finding a limit help. (Replies: 12)

  2. Help me find a limit! (Replies: 4)

  3. Help find this limit. (Replies: 3)

  4. Finding a limit help. (Replies: 4)