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Math Help - Sum of distinct product

  1. #1
    tah
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    Sum of distinct product

    Hi,

    How do I express (or approximate for larg m) the following sum in term of k,m ?

    \sum_{1\leq i_1<\cdots< i_{k} \leq m} i_1i_2\cdots i_k

    Please help,
    Thanks
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  2. #2
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    Quote Originally Posted by tah View Post
    Hi,

    How do I express (or approximate for larg m) the following sum in term of k,m ?

    \sum_{1\leq i_1<\cdots< i_{k} \leq m} i_1i_2\cdots i_k

    Please help,
    Thanks
    Do you mean you want it written like as another summation? i.e. \sum_{ i_{k} = 1} ^m i_k

    Or do you want the sequence?
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  3. #3
    tah
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    Quote Originally Posted by tsal15 View Post
    Do you mean you want it written like as another summation? i.e. \sum_{ i_{k} = 1} ^m i_k

    Or do you want the sequence?
    I want to eliminate totally the sum symbol by giving an explicit expression in term of k and m or an approximation for m large enough.
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  4. #4
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    Quote Originally Posted by tah View Post
    I want to eliminate totally the sum symbol by giving an explicit expression in term of k and m or an approximation for m large enough.
    Ok. So, for \sum_{k =1} ^m i_k,

    the sum expression is: i_1+i_2+...+ i_m

    does this help?
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  5. #5
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    And since that simple case does not have a "closed expression", the more general certainly does not.
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  6. #6
    tah
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    I mean by

    i_1i_2\cdots i_k

    a product, so the expression represent a sum of all possible products of k distinct number between 1 and m. For instance k = 2 and m = 3

    S=1\times 2 + 1\times3 + 2\times 3

    so can't we express it more simply without the sum ?

    Thanks for the helps.
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