If X is rational, there will be at least one (actually, infinitely many) Ms for which MX is an integer, and therefore the cosine is either -1 or +1

M! includes many even numbers, therefore the cosine will be +1, not -1

For irrational X, the cosine is always strictly between -1 and +1, and therefore tends to zero as n increases

(I still suspect in missing something, no pen and paper at hand)

And now I'm curious about complex X too