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Math Help - logarithmic decreasing serie

  1. #1
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    logarithmic decreasing serie

    Hi all,

    anyone knows how to compute the value of the following serie:

    alog(a) + (a-1)log(a-1)+....+(a-(a-1))log(a-(a-1))+(a-a)log(a-a)

    Many thanks in advance!
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  2. #2
    Forum Admin topsquark's Avatar
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    I'll ignore your last term, which is undefined.
    a~log(a) + (a - 1)~log(a - 1) + (a-2)~log(a - 2) +~...

    =~log(a^a) + log \left ( (a-1) ^{(a-1)} \right ) + log \left ( (a-2) ^{(a-2)} \right ) +~...

    =~log \left ( a^a \cdot (a - 1)^{(a-1)} \cdot (a - 2)^{(a - 2)} \cdot ~...~ \right )

    Take it from here.

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark View Post
    I'll ignore your last term, which is undefined.
    a~log(a) + (a - 1)~log(a - 1) + (a-2)~log(a - 2) +~...

    =~log(a^a) + log \left ( (a-1) ^{(a-1)} \right ) + log \left ( (a-2) ^{(a-2)} \right ) +~...

    =~log \left ( a^a \cdot (a - 1)^{(a-1)} \cdot (a - 2)^{(a - 2)} \cdot ~...~ \right )

    Take it from here.

    -Dan
    Thanks Dan, but now

    \prod_{i=1}^{a}(a-i+1)^{(a-i+1)}

    how do you compute this?

    Thanks again,
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