Application of the integral version of a sum I found to the zeta function

In the last post I showed this formula relating integrals and sums:

I found as an application of this formula that it can be used with a slight modification (moving the f(0) to f(1)) to investigate the Riemann sums of the form 1/(n^c) giving the zeta function like so for the case of c = 3:

This gives:

And:

And in general:

** Edit, the integrals aren't quite evaluating right so I'll have to revisit this and see if I made a typo or what...