I have been at it for a very long time. I would appreciate an early reply.
Consider the expression given at View image: Expression image .
1) Expand the expression as a series and show that the coefficient of x^(k+1) is 1^k + 2^k + 3^k + ... + (n-1)^k. (Hint: expand every e^(rx) as a series and gather like terms.)
2) Use geometric series to show that the expression can also be written as [x/(e^x -1) ]k!(e^(nx) - 1)
3) Expand both parts of the expression in 2) as series (writing the generating function in terms of the B<SUB>n, take the Cauchy product to find the coefficient of x^(k-1). Check that your answer gives the right formula.
4) Bernoulli calculated B<SUB>0 to B<SUB>11 in order to use his formula to show that:
1^10 + 2^10 + 3^10 + ... + 1000^10 = 91409924241424243424241924242500
He claimed that it took him 7.5 minutes to do this by hand. Verify that his answer is correct.