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Math Help - Calculating Sums

  1. #1
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    Calculating Sums

    Is there some sort of trick when dealing with the following sum?:
    \sum_{k=2} ^{99}  (-1)^k(k)^2

    I attempted it numerous times, but the (-1)^k is messing me up
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  2. #2
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    \sum_{k=2} ^{99} (-1)^k(k)^2 = \sum_{k=1} ^{49} ((2k)^2 - (2k+1)^2)
    = \sum_{k=1} ^{49} ((2k) - (2k+1))((2k) + (2k+1))=\sum_{k=1} ^{49} -(4k+1)
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  3. #3
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    Quote Originally Posted by quantoembryo View Post
    Is there some sort of trick when dealing with the following sum?:
    \sum_{k=2} ^{99}  (-1)^k(k)^2

    I attempted it numerous times, but the (-1)^k is messing me up
    (-1)^2\;2^2+(-1)^3\;3^2+(-1)^4\;4^2+.......

    =2^2-3^2+4^2-5^2+....

    =\displaystyle\sum_{k=1}^{49}(2k)^2-\sum_{k=1}^{49}(2k+1)^2
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    So when you are tackling these sort of problems, is it always best to set k=1?
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  5. #5
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    For this one, the theme is if you have (-1)^k (and can't simplify directly) split the sum into the positive parts and the negative parts.
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  6. #6
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    Quote Originally Posted by quantoembryo View Post
    So when you are tackling these sort of problems, is it always best to set k=1?
    If k is even

    (-1)^k=1

    If k is odd

    (-1)^k=-1

    That will simplify the situation.
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  7. #7
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    I really don't know why I am having so much trouble with this, but I just am not seeing it. Which summation formula(s) is being used?
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  8. #8
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    \sum_{k} (-1)^kf(k) = \sum_{k\, even}f(k) - \sum_{k\, odd}f(k)
    Does this make sense?
    Or are you confused about another part?
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  9. #9
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    Yes, that helps substantially. Thanks a lot!
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  10. #10
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by quantoembryo View Post
    Is there some sort of trick when dealing with the following sum?:
    \sum_{k=2} ^{99}  (-1)^k(k)^2

    I attempted it numerous times, but the (-1)^k is messing me up
    We can start from the known formula...

    \displaystyle S^{2}_{n} = \sum_{k=1}^{n} k^{2} = \frac{n\ (n+1)\ (2n+1)}{6} (1)

    ... and from (1) derive with some 'easy' step...

    \displaystyle \sum_{k=2}^{99} (-1)^{k} k^{2} = 1-S^{2}_{99} + 8\ S^{2}_{49} =- 4949 (2)

    Kind regards

    \chi \sigma
    Last edited by chisigma; January 7th 2011 at 02:25 PM.
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