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Math Help - [SOLVED] sequence of agerages

  1. #1
    Junior Member hercules's Avatar
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    [SOLVED] sequence of agerages

    Kinda' stuck,

    Let (Sn) be a nondecreasing sequence of positive real numbers and define Tn= (1/n)(s1+s2+· · ·+sn)for n ∈ N. Prove that (Tn) is a nondecreasing sequence.

    Thanks.
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  2. #2
    Junior Member hercules's Avatar
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    Just understood it....I think I got it. Thanks
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  3. #3
    Moo
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    Hi,
    Quote Originally Posted by hercules View Post
    Just understood it....I think I got it. Thanks
    Would you mind showing it please ? It can be helpful to others


    Edit : by the way, you can prove that if the sequence (Sn) converges to L, then the sequence (Tn) converges to L. This is a property of the Cesāro mean
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  4. #4
    Junior Member hercules's Avatar
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    Quote Originally Posted by Moo View Post
    Hi,

    Would you mind showing it please ? It can be helpful to others


    Edit : by the way, you can prove that if the sequence (Sn) converges to L, then the sequence (Tn) converges to L. This is a property of the Cesāro mean
    Hmm..I don't know about Cesaro mean but Sn in this case is only given as a non decreasing sequence...nothing else...so I don't know how I would proceed with that.

    Sorry, I am not good with Latex....so here is the proof informally.

    Given that Sn is a non-decreasing sequence of positive numbers, each subsequent term is greater than the previous for all 'n' in N.
    Then I wrote down several terms for the sequence Tn:
    T1 = (s1)/1
    T2 = (s1+s2)/2
    T3 = (s1+s2+s3)/3

    and noticed that T2 - T1 was positive , T3 - T2 was positive and so on...

    Since we are proving that Tn is a non-decreasing sequence as well...its every subsequent term has to be greater than the previous.
    That is,
    T(n+1)-Tn >=0


    T(n+1)-Tn = [s1+s2+...+sn+s(n+1)]/(n+1) - [s1+s2+s3+....+sn]/n

    Now to subtract make the common denominator, and get

    [n*Sn+1 -(s1+s2+s3...+sn)]/ (n*(n+1))
    This is greater than or equal to zero because we have n*Sn+1 and each is greater than each s1 thru. sn term.

    Then I used induction to show that
    T(n+1)+1 - Tn+1 is also greater than or equal to zero. Therefore the sequence must be non-decreasing.


    Hope this is understandable.
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