Bernoulli polynomials & Euler–Maclaurin summation

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April 19th 2009, 07:59 AM
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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 $1^4 + 2^4 + ... + n^4$

Thanks a lot!
April 19th 2009, 08:54 AM
chisigma

The Bernoulli polynomials are defined as...

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

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

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

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

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

Kind regards

$\chi$ $\sigma$

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