Generalization of summation of k^a

[coki2000]

Hello everybody,

Are there any generalization of this summation $$\sum_{k=1}^{n}k^{a}$$ for a>3? Thanks for your responses.

 

[fzero]

Check equ (12) on http://mathworld.wolfram.com/PowerSum.html This doesn't look like much of a simplification, but it is a sum of only a+1 terms, instead of the original n, which is presumably much larger.

 

[Gib Z]

Faulhabers Formula: http://en.wikipedia.org/wiki/Faulhaber's_formula

We can derive expressions for many series with the Euler-Maclaruin formula. Applied to this, it leads to Faulhaber's formula. It can be applied to non-integer a as well.

 

[coki2000]

Thank you for your useful helps. :)

 

[CellCoree]

Hello,

Yes, there are generalizations for this summation for values of a greater than 3. One such generalization is the Bernoulli numbers, which can be used to express the summation in terms of the Riemann zeta function. Another generalization is the Faulhaber's formula, which gives a closed form expression for the summation in terms of the Bernoulli numbers. Additionally, there are other techniques such as the Euler-Maclaurin formula and the method of finite differences that can be used to generalize this summation for various values of a. I hope this helps answer your question.