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Math Help - Help with summation

  1. #1
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    Help with summation

    Given that
    3\,\sum _{m=1}^{M} \left( \cot \left( {\frac {m\pi }{2\,M+1}} \right) <br />
 \right) ^{2}

    is a polynomial of degree r in M, find r and the coefficient of M^r.(The coefficient is an integer)

    Any ideas how to go about solving this problem?
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  2. #2
    Moo
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    Quote Originally Posted by Hweengee View Post
    Given that
    3\,\sum _{m=1}^{M} \left( \cot \left( {\frac {m\pi }{2\,M+1}} \right) <br />
 \right) ^{2}

    is a polynomial of degree r in M, find r and the coefficient of M^r.(The coefficient is an integer)

    Any ideas how to go about solving this problem?
    Using power series ?
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  3. #3
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    I have no idea. I am allowed to use MAPLE, but the command sum(expression,m=1..M) doesn't give a general solution in a polynomial form, so I am quite confused. And we haven't gotten to sequences and series in my calculus course yet.
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  4. #4
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    I have no idea how to prove it, but this formula is what you are looking for. It says \sum_{k=1}^{n-1}\cot^2(k\pi/n) = \tfrac13(n-1)(n-2). Put n=2M+1 and you get 3\sum_{k=1}^{2M}\cot^2(k\pi/(2M+1)) = 2M(2M-1).

    Notice that the sum goes from 1 to 2M. But using the fact that \cot(\pi-x)=-\cot x you can see that this is twice the sum from 1 to M.
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