Here is a way to do it. But it might be cheating, because it is basically Stirling approximation.
Instead compute the of this sequence,
.
But, by Euler-Maclaurin formula, the remainder term will happen to go to zero.
Thus, we are left with . Which means .
It appears to me all you need to do is put a bound on . Of course, Stirling approximation is the best way to go, but that would be cheating here. You can use what PaulRS did here to get good estimates on and use squeeze theorem.