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Thread: Proving order in series and sequences

  1. #1
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    Proving order in series and sequences

    Show that $\displaystyle 1^k + 2^k + 3^k + ... + (n-1)^k + n^k $ is ORDER $\displaystyle n^(k+1).$
    (where k is a positive integer)
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  2. #2
    MHF Contributor chisigma's Avatar
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    The function of x...

    $\displaystyle f(x) = x^{k} + (x-1)^{k} + (x-2)^{k} + \dots + (x-n+1)^{k}$ (1)

    ... is a polynomial of degree k and the coefficient of the term of degree k is $\displaystyle n$, i.e. is...

    $\displaystyle f(x)= n\cdot x^{k} + \dots \rightarrow f(x)=\mathcal {O} \{x^{k}\} $ (2)

    If we set in (2) $\displaystyle x=n$ we have...

    $\displaystyle f(n)= n^{k+1} + \dots \rightarrow f(n)=\mathcal {O} \{n^{k+1}\} $ (3)

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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