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Math Help - Double Sigma Notation Help

  1. #1
    kap
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    Double Sigma Notation Help

    Hi guys,

    I am having a little trouble understanding a multi sigma notation. I have a double sigma notation, the first sigma is directly followed by a second sigma with no arithmetic operator between them. i am given:

    S(n) = (First Sigma) (Second Sigma) i

    [the "i" is the formula after the second sigma. indexes : first sigma==>lower index :: k=1, upper index=n
    second sigma==> lower index:: i=1, upper index=k]

    i am asked to prove that

    S(n) =[ n(n + 1)(n + 2)]/6

    (I don't know how to make the sigma signs appear in this forum.)

    I am asking if someone can explain to me what the double sigma notation means.

    My understanding is this:
    I replace the second sigma with a formula, so that the formula preceded by the first sigma is used for the proof .

    Is that right?

    Need help.



    Thanks.
    Last edited by kap; March 27th 2010 at 04:11 PM. Reason: made a mistake
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  2. #2
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    Quote Originally Posted by kap View Post
    Hi guys,

    I am having a little trouble understanding a multi sigma notation. I have a double sigma notation, the first sigma is directly followed by a second sigma with no arithmetic operator between them. i am given:

    S(n) = (First Sigma) (Second Sigma) i

    [the "i" is the formula after the second sigma. indexes : first sigma==>lower index :: k=1, upper index=n
    second sigma==> lower index:: i=1, upper index=k]


    So this seems to be S(n)=\sum^n_{k=1}\sum^k_{i=1}i =1+(1+2)+(1+2+3)+\ldots +(1+2+\ldots+ n).

    Now, use that for any k\in\mathbb{N}\,,\,\,1+2+\ldots +k=\frac{k(k+1)}{2} and 1^2+2^2+\ldots +k^2=\frac{n(n+1)(2n+1)}{6} (you can easily prove both formulae by

    induction), and S(n)=\sum^n_{k=1}\frac{k(k+1)}{2}=\frac{1}{2}\left  (\sum^n_{k=1}k^2+\sum^n_{k=1}k\right) , plus a little algebra to deduce the result they want.

    The double notation may be understood as follows: make the first index run, and for each value it takes make the second sum's index run.

    Tonio


    i am asked to prove that

    S(n) =[ n(n + 1)(n + 2)]/6

    (I don't know how to make the sigma signs appear in this forum.)

    I am asking if someone can explain to me what the double sigma notation means.

    My understanding is this:
    I replace the second sigma with a formula, so that the formula preceded by the first sigma is used for the proof .

    Is that right?

    Need help.



    Thanks.
    .
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  3. #3
    kap
    kap is offline
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    Thank you very much for your reply
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