x:=1/n, y:=(1/(n+1))*(1-1/n)^n=(x/(x+1))*(1-x)^(1/x)
As n->+∞, x->+0, y/x->1/e, y->+0

log(A[n]^n)
=(log(1-x))/x+(log(1-y))/x
=(log(1-x))/x+(y/x)*(log(1-y))/y
->-1-1/e (n->+∞)　　[lim[t->0]((log(1-t))/t)=-1]

A[n]^n=exp(log(A[n]^n)->e^(-1-1/e) (n->+∞)