No result for a summation in V12

Questions
1

I tried to solve a summation with Mathematica 12 but seems it can’t solve it. Can somebody help me?

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2

Do you have a reason to expect that the summation is analytically computable?
Have you tried some simple cases with specific numerical values for some of the parameters?

3

it Converges for 0<k<1. im not sure if there is a analytical solution for it.

4

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.