If I split the sum up, I get only part way:
Sum[j^(-a - 1), {j, i + 1, n}, Assumptions -> 0 < a < 1]
Sum[%, {i, n}, Assumptions -> 0 < a < 1]
(*
HurwitzZeta[1 + a, 1 + i] - HurwitzZeta[1 + a, 1 + n]
Sum[HurwitzZeta[1 + a, 1 + i] - HurwitzZeta[1 + a, 1 + n], {i, n},
Assumptions -> 0 < a < 1]
*)
And one half-step further:
Distribute[%]
(*
Sum[HurwitzZeta[1 + a, 1 + i], {i, n}, Assumptions -> 0 < a < 1] -
n Zeta[1 + a, 1 + n]
*)
And that's as far as I can take it.
