The limes you wrote in front does not make sense, as the result of the series would not include any variable n!
The series itself (without the lim) diverges, but Mathcads symbolics is not powerful enough to come to that conclusion - it would require comparisons with well know similar diverging series, which Mathcad obviously has not built in.
The limes (without the series) converges against 0, but of course you won't need Mathcad to see that.
Whether matchad can show it or not, the series diverges. The nth term is always greater than (1/2)*n^(-1/2), which diverges. The latter term n^(-p), without the factor 1/2, is the general term of a series for the Riemann zeta function, which diverges for p <=1, and converges for p>1.
Mathcad symbolics usually need a helping hand, as has been pointed out many times. In mcd11, the symbolics got two different answers for the referecne series - one correct, one not - with a seemingly innocuous change.
Its a shame that newer Mathcad versions with Mupad as symbolic engine have still more problems with that rather simple limes. Only the float-mode (how should we call that mode which is automatically taken if MC symbolics encounters even a single decimal point - "semi-exact"?) it returns the right answer.
That "float-trick" however does not work for the original limit:
