Given: a_n = (n!)/n^n
Then: a_(n+1) = ((n + 1)!)/(n + 1)^(n + 1)
Therefore:
a_(n+1)/a_n = (((n + 1)!)/(n + 1)^(n + 1))/((n!)/n^n)
a_(n+1)/a_n = ((n + 1)!n^n)/(n!(n + 1)^(n + 1))
a_(n+1)/a_n = ((n + 1)!)/(n!) xx n^n/((n + 1)^(n + 1))
a_(n+1)/a_n = (n!(n + 1))/(n!) xx n^n/((n + 1)^(n + 1))
a_(n+1)/a_n = (color(red)(cancel(color(black)(n!)))(n + 1))/color(red)(cancel(color(black)(n!))) xx n^n/((n + 1)^(n + 1))
a_(n+1)/a_n = (n + 1) xx n^n/((n + 1)^(n + 1))
a_(n+1)/a_n = color(red)(cancel(color(black)((n + 1)))) xx n^n/((n + 1)^(n color(red)(cancel(color(black)(+ 1)))))
a_(n+1)/a_n = n^n/((n + 1)^n
a_(n+1)/a_n = (n/(n + 1))^n