General formula for sum of k-th powers

[bomba923]

Quite Curious

$$\begin{gathered}<br /> \sum\limits_{i = 1}^n i = \frac{{n\left( {n + 1} \right)}}<br /> {2} \hfill \\<br /> \sum\limits_{i = 1}^n {i^2 } = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}<br /> {6} \hfill \\<br /> \sum\limits_{i = 1}^n {i^3 } = \frac{{n^2 \left( {n + 1} \right)^2 }}<br /> {4} \hfill \\<br /> \vdots \hfill \\<br /> \left( {etc} \right) \hfill \\ <br /> \end{gathered}$$
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But in general,

$$\forall k \in \mathbb{N} ,$$

what is the general summation formula for

$$\sum\limits_{i = 1}^n {i^k } \; {?}$$

 

[shmoe]

You can get something in general in terms of Bernoulli numbers:

http://en.wikipedia.org/wiki/Bernoulli_number

Mathworld gives it explicitly up to the sum of 10th powers:

http://mathworld.wolfram.com/PowerSum.html