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Math Help - Power rule.

  1. #1
    Newbie BraveHeart's Avatar
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    Power rule.

    Prove that:

    \displaystyle{\left(\sum_{k=0}^{n}a_{k}x^{k}\right  )' = \sum_{k=1}^{n}ka_{k}x^{k-1}}

    and/or that

    \int\left(\sum_{k=0}^{n}a_{k}x^{k}\right){dx} = \sum_{k=0}^{n}\frac{a_{k}x^{k+1}}{k+1}+c
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  2. #2
    MHF Contributor
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    I'm feeling an induction proof coming on. It's not like they are infinite sums. You won't have to worry about convergence.

    Maybe I'm wrong.
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  3. #3
    MHF Contributor

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    No, completely right. Induction on n and the "sum rule": (f(x)+ g(x))'= f'(x)+ g'(x).
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  4. #4
    Newbie BraveHeart's Avatar
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    I'm not sure how to proceed with induction.

    Because \left(\sum_{k=0}^{n}a_{k}x^{k}\right)' = \sum_{k=0}^{n}\left(a_{k}x^{k}\right)' = \sum_{k=0}^{n}a_{k}\left(x^{k}\right)', what needs to be proven is that \left(x^{k}\right)' = kx^{k-1}. The same goes to \int\left(\sum_{k=0}^{n}a_{k}x^{k}\right){dx}<br />
.
    Last edited by BraveHeart; February 10th 2010 at 09:35 PM.
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