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Thread: Bernoulli polynomials & Euler–Maclaurin summation

  1. #1
    Member Last_Singularity's Avatar
    Dec 2008

    Bernoulli polynomials & Euler–Maclaurin summation

    Could you please point me in the right direction for this problem?

    Question: Use either Bernoulli polynomials or the Euler–Maclaurin summation formula to evaluate $\displaystyle 1^4 + 2^4 + ... + n^4$

    Thanks a lot!
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  2. #2
    MHF Contributor chisigma's Avatar
    Mar 2009
    near Piacenza (Italy)
    The Bernoulli polynomials are defined as...

    $\displaystyle B_{k} (x) = \sum_{i=0}^{k} \binom {k}{i}\cdot B_{k-i}\cdot x^{i}$

    where the $\displaystyle B_{i}$ arer the 'Bernoulli numbers' defined in ricorsive manner as...

    $\displaystyle B_{0}=1$ , $\displaystyle \sum_{i=0}^{k-1}\binom {k}{i}\cdot B_{i} =0$

    The Bernoulli polynomial are useful [among other things...] for the following formula...

    $\displaystyle \sum_{i=1}^{n} i^{k} = \frac {B_{k+1}(n+1) - B_{k+1}(0)}{k+1}$

    Kind regards

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