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1^k+2^k+...+n^k= Sum(j=0, k)[(k choose j)Bj*((n+1)^(k+1-j)/(k+1-j))

where Bj is the jth Bernoulli number, and k=1,2,...

We are given

1+e^z+e^(2z)+...+e^(nz)= ((e^((n+1)z)-1)/z)(z/(e^z-1))

and told to write each side as a power series to derive the top.

So far I have proven,

1+e^z+e^(2z)+...+e^(nz)= Sum(m=0, infinity)[(0^m+1^m+...+n^m)/m!] and

((e^((n+1)z)-1)/z)(z/(e^z-1))= Sum(j=0, k)[(k choose j)Bj*((n+1)^(k+1)/(k+1-j))

But now I am stuck, where do I go from here?

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# Application of Bernoulli numbers

Can you offer guidance or do you also need help?

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