Results 1 to 3 of 3

Thread: a_{n+1}=a_n+1/S_n - sequence, limit

  1. #1
    Senior Member
    Joined
    Nov 2007
    Posts
    329

    a_{n+1}=a_n+1/S_n - sequence, limit

    $\displaystyle a_1=1$, $\displaystyle a_{n+1}=a_n+\frac 1{S_n}$ where $\displaystyle n=1,\,2,\ldots$ and $\displaystyle S_n=a_1+a_2+\ldots +a_n$. Prove that $\displaystyle \lim_{n\rightarrow\infty}a_n=\infty$. Take $\displaystyle b_n=n(a_n^2-a_{n-1}^2)$; does it have a limit if yes what?
    Last edited by james_bond; Oct 9th 2009 at 09:16 AM. Reason: Corrected an error in the statement.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1
    Quote Originally Posted by james_bond View Post
    $\displaystyle a_1=1$, $\displaystyle a_{n+1}=a_n+\frac 1{S_n}$ where $\displaystyle n=1,\,2,\ldots$ and $\displaystyle S_n=a_1+a_2+\ldots +a_n$. Prove that $\displaystyle \lim_{n\rightarrow\infty}a_n=0$.
    That is impossible. $\displaystyle \left(a_n\right)$ is clearly increasing and each term is $\displaystyle \ge 1$.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2007
    Posts
    329
    Quote Originally Posted by Plato View Post
    That is impossible. $\displaystyle \left(a_n\right)$ is clearly increasing and each term is $\displaystyle \ge 1$.
    I'm sorry. My bad. Corrected it!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. limit of sequence
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Jun 28th 2011, 05:09 PM
  2. Replies: 2
    Last Post: Oct 26th 2010, 10:23 AM
  3. limit of a sequence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Mar 24th 2010, 05:08 PM
  4. Limit of a Sequence
    Posted in the Differential Geometry Forum
    Replies: 30
    Last Post: Mar 6th 2010, 09:10 AM
  5. Limit of sequence
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Mar 12th 2008, 03:16 PM

Search Tags


/mathhelpforum @mathhelpforum