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