The summer after high school I found the formula: Sum_{k=1..n} 1/k^s = (1/Gamma(s))*Integral_{0..1} ((1-x^n)/(1-x))* ln(1/x)^(s-1) dx [if I’m remembering correctly], which can be extended on the RHS to at least any real s > 1 and probably lots more complex s as well. (As I’ve mentioned, I showed this to my freshman advisor, who reached over and pointed out the same formula in his Whittaker & Watson.) —Dan On Mar 3, 2014, at 2:42 PM, Warren D Smith <warren.wds@gmail.com> wrote:
H(n) = SUM(1/j, j=1..n) is the nth "harmonic number."
QUESTION: what if we want to generalize this to real or complex n, not merely integer n, similarly to how Euler converted factorial to gamma function? . . . Euler also had found the integral representation
H(x) = INTEGRAL( (1 - t^x) / (1-t), t=0..1 )
which is equivalent to this.