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The sum:

$$S_m(n) = 1^m + 2^m + 3^m + 4^m + 5^m...+ n^m$$

Can be calculated by this formula, called the "Bernoulli formula" in wikipedia

$$S_m(n) = \frac{1}{m+1}\sum_{k=0}^m {m+1\choose k}B_k n^{m+1-k} $$

I tried to find an intuitive proof of this formula (not by induction or anything) but I didn't find any. Does somebody knows any proof or can help me with some link?

Thank you so much!

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What is "an intuitive proof"? The proof relies on some properties of the Bernoulli polynomials, which are not too hard to prove with appropriate defintions. – Pedro Tamaroff May 7 '13 at 1:10
I mean 'intuitive' proof is not a proof by induction (what doesn't acrescent any knowledge to me) and not a proof that "disassembles" the formula to a result that proves itself. I mean 'intuitive' because it shows how the author of the formula did it, intuitively – Lucas Zanella May 7 '13 at 1:17
Does the proof in help? – lhf May 7 '13 at 1:19
I think this link can help you… – mhd.math May 19 '13 at 11:20

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