Question

prove to me sum of (n!)/(n)^(n) converge or diverge

Answer 1

\[f(n)=\frac{n!}{n^n}\]lets find some values

\[f(1)=\frac{1!}{1^1}=1\]
\[f(2)=\frac{2!}{2^2}=2/4=1/2=0.5\]
\[f(3)=\frac{3!}{3^3}=6/27=\dot2\]
\[f(4)=\frac{4!}{4^4}=12/256=0.046875\]
\[f(10)=\frac{10!}{10^{10}}=3628800/10000000000\approx0.000363\],


as the input increases the output decreases