Results 1 to 2 of 2

Math Help - Cesaro Sums (Real Analysis)

  1. #1
    Newbie
    Joined
    Jan 2013
    From
    Rochester
    Posts
    1

    Cesaro Sums (Real Analysis)

    We have defined the convergence of a series,

    SUM [aj] from j=1 to infinity

    in terms of the sequence of partial sums associated with that series,

    Sn = SUM [aj] from j=1 to n

    We say that the series converges if, and only if, the sequence of its partial sums converges.

    The mathematician Ernesto Cesaro introduced another idea to this field, an idea that we call Cesaro summability. We define another sequence,

    Pm=(1/m) * SUM[Sn] from n=1 to m

    That is, Pm is the arithmetic mean of the first m partial sums. If the sequence {Pm} converges to z, we say that SUM[aj] from j=1 to infinity is a Cesaro summable and that its Cesaro sum is z.



    1) Prove that if a series converges, then it is Cesaro summable and that the series converges to its Cesaro sum.

    2) Prove that if a series of positive terms diverges it is not Cesaro summable.




    Guys, I'm really stuck on the above problem. Any tip or guidance would be much appreciated. Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,619
    Thanks
    592

    Re: Cesaro Sums (Real Analysis)

    Hey JamesTrack.

    Have you looked at the proofs that involve Cauchy-Sequences? Maybe you could adapt those to the question that you need to answer.

    It was a while back, but I do recall the proofs of Cauchy-Sequences being used for all kinds of convergence proofs of partial sums.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. cesaro summable proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 15th 2010, 06:43 PM
  2. convergence of cesaro averages
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 22nd 2010, 01:02 AM
  3. Real Analysis
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: January 30th 2010, 08:14 AM
  4. Cesaro Sum Help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 5th 2009, 06:54 PM
  5. Real Analysis
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 27th 2009, 06:02 AM

Search Tags


/mathhelpforum @mathhelpforum