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!

Analysis question

  1. Oct 16, 2006 #1
    X = (xn) is a sequence of strictly positive real numbers, where (xn) is x subscript n, such that lim(x(n+1)/xn) < 1. Show that for some r with 0<r<1 and some C>0 that 0<xn<Cr^n for all sufficiently large natural numbers n. and that lim(xn) = 0

    So for I have this:
    choose r such that lim(x(n+1)/xn)<r<1 and take a neighborhood of this limit to be the interval (-1,r) So there exists a natural number K such that 0<x(n+1)/xn<r for all n>=K. I can also write r = 1/(1+a) where a>0 and show that lim(r^n)=0. All I need to show now is that xn<Cr^n. Because I know that if lim(r^n)=0 and ||xn - 0||<=C|r^n| where C>0 then lim(r^n)=0. I'm not really sure how to get the xn<Cr^n though. Any help or suggestions?
  2. jcsd
  3. Oct 16, 2006 #2


    User Avatar
    Homework Helper

    You've shown xn+1/xn<r for n>=K. You can rewrite this as xn+1<r*xn. Apply induction to get an inequality involving xk+n and xk.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Analysis question
  1. Analysis question (Replies: 10)

  2. Analysis question (Replies: 0)

  3. Analysis question (Replies: 1)

  4. An analysis question (Replies: 20)

  5. An analysis question (Replies: 4)