In using **Root test** on a series sum a_n, we determine 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...

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In using **Root test** on a series sum a_n, we determine 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 (-1)^(n-1) *(3/2)^n/n^2` .

`lim_(n-gtoo) |((-1)^(n-1) *(3/2)^n/n^2 )^(1/n)| =lim_(n-gtoo) |(-1)^((n-1)*1/n)(3/2)^(n*1/n)/n^(2*1/n)|`

`=lim_(n-gtoo) |(-1)^(n/n-1/n)(3/2)^(n/n)/n^(2/n)|`

`=lim_(n-gtoo)( 1 * (3/2)^1/n^(2/n))`

`=lim_(n-gtoo) (3/2)/n^(2/n)`

Note: `|(-1)^(n/n-1/n)| = 1`

Apply the limit property: `lim_(x-gta)[(f(x))/(g(x))] =(lim_(x-gta) f(x))/(lim_(x-gta) g(x)).`

`lim_(n-gtoo) (3/2)/n^(2/n)=(lim_(n-gtoo) 3/2)/(lim_(n-gtoo)n^(2/n))`

`= ((3/2))/1`

` =3/2 or 1.5`

The limit value `L = 3/2 or 1.5` satisfies the condition: `Lgt1` since `3/2gt 1` or `1.5gt1` .

Thus, the series `sum_(n=1)^oo (-1)^(n-1) *(3/2)^n/n^2` is **divergent**.