`sum_(n=1)^oo ((-3n)/(2n+1))^(3n)` Use the Root Test to determine the convergence or divergence of the series.

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To apply 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.

In order to apply Root Test in determining the convergence or divergence of the series `sum_(n=1)^oo ((-3n)/(2n+1))^(3n)` , we let:

`a_n =((-3n)/(2n+1))^(3n)`

We set-up the limit as: 

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

Note: `| - f(n)| = f(n)` .

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

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

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

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

Apply the limit property: `lim_(x-gta)[f(x)]^b =[lim_(x-gta)f(x)]^b` such that `b` is a real constant.

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

To evaluate the limit `lim_(n-gtoo) (3n)/(2n+1)` , we divide each term by the highest denominator  power: n.

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

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

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

                   ` =3/(2+1/oo)`

                   ` =3/(2+0)`

                   ` =3/2`

Applying `lim_(n-gtoo) (3n)/(2n+1)=3/2` , we get:

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

Apply Law of Exponent: `(x/y)^n = x^n/y^n ` and simplify.

`(3/2)^3 =3^3/2^3`

         ` = 27/8 or 3.375`

The limit value `L = 27/8 or 3.375` satisfies the condition: `Lgt1` .

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

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