I need to prove(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Application of Bernoulli numbers

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**