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Is it any easier than computing $\sum^N_{i=0}i^k$?

I've came to such a sum while trying to solve a task from a basic course on probability... If $\frac{\sum^N_{i=0}i^{k+1}}{\sum^N_{i=0}i^k}$ can't be easily solved, then I think I'll share this task and my solution attempt with You...

gaazkam
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    See whether this helps you: https://www.google.co.in/url?sa=t&source=web&rct=j&url=http://www.math.tulane.edu/~vhm/sum-powers.pdf&ved=0ahUKEwi0-qGP-bvYAhXHsI8KHUxEDa4QFgg1MAU&usg=AOvVaw23j2Nbas6MieFDapswQCRO –  Jan 03 '18 at 13:57
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    See here also: http://mathworld.wolfram.com/PowerSum.html –  Jan 03 '18 at 13:57

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