[seqfan] Sum of expression with harmonic numbers should simplify to Zeta?

[Jean-François Alcover]

Dear Seqfans,

Thanks in advance to the (kind) specialist
who could explain how can be proved that this expression:

Sum[ HarmonicNumber[k]/k^n - (PolyGamma[k+1] + EulerGamma)/(k+1)^n, {k, 1,
Infinity}]

always simplifies to Zeta[n+1].

J.-F. Alcover

P.s.
FullSimplify returns:
Sum[ (-(k*(1+k))^(-n))*(k^n - (1+k)^n)*HarmonicNumber[k], {k, 1, Infinity}]