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While @Szabolcs has provided the correct work-around, the actual issue has to do with automatic use of Compile for function evaluation. The affected range is such that the value of Gamma[1+n] is still a machine real, but their product is out-of-bound. The issue comes, because the default setting of Compile's RuntimeOptions is to tolerate machine arithmetic ...


Gamma is an extremely quickly increasing function, so you're dealing with the ratio of huge numbers here. Something similar to catastrophic cancellation can happen. Fortunately, Mathematica is very good at dealing with this situation if you let it use arbitrary precision instead of machine precision. Change 0.5 to 1/2 and add something like ...

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