How can we solve sums of powers of integers using differences and integrals?

[eljose]

Let,s suppose we want to do this sum:

$$1+2^{m}+3^{m}+...+n^{m}$$ n finite

then we could use the property of the differences:

$$\sum_{n=0}^{n}(y(k)-y(k-1))=y(n)-y(0)$$


so for any function of the form f(x)=x^{m} m integer you need to solve:

$$y(n)-y(n-1)=n^{m}$$ i don,t know how to solve

it.. i have tried the ansatz y(n)=K(n) with K(n) a Polynomial of degree m+1 but i don,t get the usual results for the sum..could someone help?..

 

[fresh_42]

It helps sometimes to write the summands as integrals, e.g. like here
https://www.physicsforums.com/threads/micromass-big-integral-challenge.867904/page-2#post-5451293
in which case you can get to the related gamma function.