hn[k_, r_] := Inactive[Sum][1/n^r, {n, 1, k}]
hn[k, r]

When Activate
'd this is the HarmonicNumber
hn[k, r] // Activate
(* HarmonicNumber[k, r] *)
For your specific example
hn[k, -1/2]

hn[k, -1/2] // Activate
(* HarmonicNumber[k, -(1/2)] *)
To see the expanded form for the first several cases
Table[hn[k, -1/2] // Activate, {k, 1, 7}]
(* {1, 1 + Sqrt[2], 1 + Sqrt[2] + Sqrt[3], 3 + Sqrt[2] + Sqrt[3],
3 + Sqrt[2] + Sqrt[3] + Sqrt[5], 3 + Sqrt[2] + Sqrt[3] + Sqrt[5] + Sqrt[6],
3 + Sqrt[2] + Sqrt[3] + Sqrt[5] + Sqrt[6] + Sqrt[7]} *)
or
Table[hn[k, -1/2], {k, 1, 7}] // Activate
(* {1, 1 + Sqrt[2], 1 + Sqrt[2] + Sqrt[3], 3 + Sqrt[2] + Sqrt[3],
3 + Sqrt[2] + Sqrt[3] + Sqrt[5], 3 + Sqrt[2] + Sqrt[3] + Sqrt[5] + Sqrt[6],
3 + Sqrt[2] + Sqrt[3] + Sqrt[5] + Sqrt[6] + Sqrt[7]} *)
If activated prior to summation, HarmonicNumber
appears for higher values of k
Table[Evaluate[hn[k, -1/2] // Activate], {k, 1, 7}]
(* {1, 1 + Sqrt[2], 1 + Sqrt[2] + Sqrt[3], 3 + Sqrt[2] + Sqrt[3],
HarmonicNumber[5, -(1/2)], HarmonicNumber[6, -(1/2)],
HarmonicNumber[7, -(1/2)]} *)
FunctionExpand
will convert to a sum
% // FunctionExpand
(* {1, 1 + Sqrt[2], 1 + Sqrt[2] + Sqrt[3], 3 + Sqrt[2] + Sqrt[3],
3 + Sqrt[2] + Sqrt[3] + Sqrt[5], 3 + Sqrt[2] + Sqrt[3] + Sqrt[5] + Sqrt[6],
3 + Sqrt[2] + Sqrt[3] + Sqrt[5] + Sqrt[6] + Sqrt[7]} *)