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We know $\sum_{0}^{N}m^2=\frac{N(N+1)(2N+1)}{6}$.

Is there a generic expression for $\sum_{0}^{N}m^n$ where $n$ is an even number?

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There is actually an expression for that sum when $n$ is any natural number in terms of the Bernoulli numbers. In particular it will hold when $n$ is an even natural number. http://en.wikipedia.org/wiki/Faulhaber%27s_formula

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  • $\begingroup$ You're very welcome. $\endgroup$ – user71352 May 31 '14 at 20:56

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