Thanks to both of you.

I already proved that the limit was exp(pi/8) but I was curious about the capacities of mathematica to find it. I didn't know either the NLimit command which seems very useful.

Thank you very much for taking the time to answer me

This is really a math question. Mathematica simply cannot do the limit.

First an approximation.

partials = Table[Product[1 + Sqrt[k (n - k)]/n^2, {k, 1, n}], {n, 20, 400, 20}];
N[partials]

(* Out[470]= {1.4681, 1.47549, 1.47761, 1.47858, 1.47913, 1.47948, \
1.47972, 1.4799, 1.48003, 1.48014, 1.48022, 1.48029, 1.48035, 1.4804, \
1.48044, 1.48048, 1.48051, 1.48054, 1.48057, 1.48059} *)

We can also get the limiting value numerically like so.

NLimit[Product[1 + Sqrt[k (n - k)]/n^2, {k, 1, n}], n -> Infinity]

(* Out[472]= 1.48097 *)

This happens to be a really good approximation to Exp[Pi/8]. There are a few steps required to show that that is in fact the limit.