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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Search for Bernoulli numbers. – Anurag A May 31 '14 at 19:03
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$\sum_{m=1}^N m^n=\frac{(N+1+B)^{n+1}-B^{n+1}}{n+1}$. See also http://math.stackexchange.com/questions/642625/how-is-the-bernoulli-numbers-for-example-as-against-b-2/642689#642689 to find the value of $B^n\Doteq B_n$ for any arbitrary $n$. – user91500 May 31 '14 at 19:10
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Possible duplicate of How can I compute the sum of this formula $ \displaystyle\sum_{i=1}^{100} i^8- 2 i^2 $ – user91500 Mar 10 '16 at 08:35
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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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