Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By FernandoRevilla

Math Help - Convergence of a positive series - true or false question

  1. #1
    Junior Member
    Joined
    Jun 2007
    Posts
    62

    Convergence of a positive series - true or false question

    True/False:
    If a positive series \Sigma a_n converges then it must be that \lim_{n\to \infty} \sqrt[n]{a_n} \leq 1.

    According to the textbook the answer is false, but I can't find a counterexample. Any ideas?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Convergence of a positive series - true or false question

    Choose

    a_n=\begin{Bmatrix} 0 & \mbox{ if }& n\mbox{ even}\\\dfrac{1}{n^2} & \mbox{if}& n\mbox{ odd}\end{matrix}

    then, \sum_{n=0}^{+\infty}a_n is convergent (compare with \sum_0^{+\infty}1/n^2), however does not exist \lim_{n\to +\infty}\sqrt[n]{a_n}.
    Thanks from loui1410
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: October 5th 2011, 01:45 PM
  2. Convergence True or False Proof
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 26th 2010, 06:38 PM
  3. True or False. Prove if true show counter example if false.
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: March 2nd 2010, 11:54 AM
  4. Replies: 3
    Last Post: May 15th 2009, 12:54 PM
  5. Series True/False prob
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 19th 2007, 09:59 PM

Search Tags


/mathhelpforum @mathhelpforum