The basic approximation I use is n! ~ (n/e)^n so ln(n!) ~ n(ln(n)-1). The basic inequality is (n/e)^n < n! < (n/e)^{n+1}. When considering convergence of series

The basic approximation I use is n! ~ (n/e)^n so ln(n!) ~ n(ln(n)-1). The basic inequality is (n/e)^n < n! < (n/e)^{n+1}. When considering convergence of series using the n-th root test, (n!)^{1/n} ~ n/e so ((an)!)^{1/n} = (((an)!)^{1/(an)})^a ~(an/e)^a = a^an^a/e^a.