Find a, b, c

[o.m.]

Let f be a real function defined on [0;1] which is \( C^3 \) (it means that f''' there exists and it is continous).

Find a, b, c such that

\( \frac{1}{n}\sum_{k=1}^{n}f(k/n)=a+\frac{b}{n}+\frac{c}{n^2}+\frac{u(n)}{n^2} \)

with u(n) tend to zero when n tend \( \infty \).

[aleph]

This is just a special case of the Euler-Maclaurin summation formula.