How do you use the Root Test on the series sum_(n=1)^oo((n!)/n)^n ?

1 Answer
Sep 22, 2014

By Root Test,

sum_{n=1}^infty({n!}/{n})^n diverges.

Let us look at some details.

Let a_n=({n!}/n)^n=[(n-1)!]^n.

Consider:

lim_{n to infty}rootn{|a_n|}=lim_{n to infty}rootn{[(n-1)!]^n} =lim_{n to infty}(n-1)! =infty,

which is greater than 1; therefore, we can conclude that the series diverges by Root Test.