Results 1 to 2 of 2

Math Help - quick limit

  1. #1
    Junior Member
    Joined
    Sep 2007
    Posts
    38

    quick limit

    I have to figure out this limit:
    Let (an) be a sequence whose limit as n approaches infinity is 1. Suppose that an does not equal 1 for all n in the natural numbers. For any fixed k in the natural numbers, find the limit as n approaches infinity of
    an + (an)^2 + (an)^3 + ..... + (an)^k - k
    ---------------------------------------
    an - 1.

    I know the limit is 1+2+3+.........+k. Yet I can't solve it algebraically. It seems so many of the individual limits go to infinity. Can someone show me how to do this? Thanks. I really appreciate it.
    Last edited by BrainMan; September 30th 2007 at 07:35 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    \displaystyle\lim_{n\to\infty}\frac{a_n+a_n^2+a_n^  3+\ldots+a_n^k-k}{a_n-1}=
    \displaystyle=\lim_{n\to\infty}\frac{a_n-1}{a_n-1}+\lim_{n\to\infty}\frac{a_n^2-1}{a_n-1}+\lim_{n\to\infty}\frac{a_n^3-1}{a_n-1}+\ldots+\lim_{n\to\infty}\frac{a_n^k-1}{a_n-1}=
    \displaystyle=1+\lim_{n\to\infty}(a_n+1)+\lim_{n\t  o\infty}(a_n^2+a_n+1)+\ldots+\lim_{n\to\infty}(a_n  ^{k-1}+a_n^{k-2}+\ldots+a_n+1)=
    \displaystyle=1+2+3+\ldots+k=\frac{k(k+1)}{2}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Quick limit question
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 1st 2011, 03:04 PM
  2. Quick limit question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 26th 2010, 01:13 AM
  3. Quick Limit Question
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 2nd 2009, 11:56 PM
  4. Quick limit help.
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 23rd 2008, 10:58 PM
  5. Quick question on a limit
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 16th 2008, 08:38 PM

Search Tags


/mathhelpforum @mathhelpforum