[REOPENED]
A formula for the power sums: $1^n+2^n+\dotsc +k^n=\,$?A formula for the power sums: $1^n+2^n+\dotsc +k^n=\,$? , from a user investigating that sum, presents a method that the OP tried to get the sum in closed form, observes that it doesn't work well, and then asks “are there other ways?”
The purportedly "duplicate" question Finite Sum of Power?Finite Sum of Power? also asks for the closed form, but not for a derivation, or for techniques to approach the problem, and the answer there doesn't give any.
(I voted to close the question, and on reflection, I regret my vote. I have voted to reopen it. I will try to be more careful in the future.)