Question

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

Answer 1

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.