Recall the **Root test** determines the limit as:

`lim_(n-gtoo) root(n)(|a_n|)= L`

or

`lim_(n-gtoo) |a_n|^(1/n)= L`

Then, we follow the conditions:

a) `Llt1` then the series is **absolutely convergent**

b) `Lgt1` then the series is **divergent**.

c) `L=1` or *does not exist* then the **test is inconclusive**. The series may be divergent, conditionally convergent, or absolutely convergent.

We may apply the Root Test to determine the convergence or divergence of the series `sum_(n=1)^oo(n/500)^n` .

For the given series `sum_(n=1)^oo(n/500)^n` , we have `a_n = (n/500)^n.`

Applying the Root test, we set-up the limit as:

`lim_(n-gtoo) |(n/500)^n|^(1/n)=lim_(n-gtoo) ((n/500)^n)^(1/n)`

Apply Law of Exponent: `(x^n)^m = x^(n*m)` .

`lim_(n-gtoo) ((n/500)^n)^(1/n) =lim_(n-gtoo) (n/500)^(n*(1/n))`

`=lim_(n-gtoo) (n/500)^(n/n)`

`=lim_(n-gtoo) (n/500)^1`

`=lim_(n-gtoo) (n/500)`

Evaluate the limit as `n` approaches `oo` .

`lim_(n-gtoo) (n/500) =1/500lim_(n-gtoo) n`

`= 1/500 * oo`

`=oo`

The limit value `L =oo` satisfies the condition: `Lgt1` .

Thus, the series `sum_(n=1)^oo(n/500)^n` is **divergent**.