User blog comment:King2218/Count to "infinity"!/@comment-3427444-20140927010624/@comment-1605058-20141006140041

Stirling's approximation is actually asymptotically correct, so that, when we get to large numbers, the error is really negligible. If I'm not mistaken, this approximation is off of the correct value by factor of 1,1 or less, always.