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 ((2n)/(n+1))^n` .

For the given series `sum_(n=1)^oo ((2n)/(n+1))^n` , we have `a_n =((2n)/(n+1))^n.`

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

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

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

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

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

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

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

Evaluate the limit.

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

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

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

` = 2 /(1 +1/oo)`

` = 2 /(1 +0)`

` = 2/1`

` =2`

The limit value `L =2` satisfies the condition: `Lgt1` .

Conclusion: The series `sum_(n=1)^oo ((2n)/(n+1))^n` is
**divergent**.

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