t[n_, k_] := (-1)^(n - k) Binomial[n, k]

Sum[t[n, k]/k, {k, 1, n}] == (-1)^(n + 1) Sum[
    1/k, {k, 1, n}] = (-1)^(n + 1) HarmonicNumber[n] // FullSimplify

(* True *)