Try with numerical values for the parameters, for example
With[{\[Omega] = 0, k = 1/2, A = 1},
Sum[Sqrt[(E^(-(2 + n) \[Omega]/T) (-1 + E^(\[Omega]/T))^2 +
A k^n n)^2 + 4 A^2 k^(2 n)], {n, Infinity}]]
If that does not give a closed form, then the general case is hopeless too.
