To apply the **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 (n/(2n+1))^n` .

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

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

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

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

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

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

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

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

Evaluate the limit.

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

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

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

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

` = 1 /(2 +0)`

` = 1/2`

The limit value `L =1/2` satisfies the condition: `Llt1` .

**Conclusion:** The series `sum_(n=1)^oo(n/(2n+1))^n` converges absolutely

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