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`sum_(n=1)^oo n(7/8)^n` Use the Root Test to determine the convergence or divergence of the series.

To apply the Root test on a series `sum a_n` , we determine a 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 the Root Test in determining the convergence or divergence of the series `sum_(n=1)^oo n(7/8)^n` , we let : `a_n =n(7/8)^n` .


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


`lim_(n-gtoo) |n(7/8)^n|^(1/n) =lim_(n-gtoo) (n(7/8)^n)^(1/n)`


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


`lim_(n-gtoo) (n(7/8)^n)^(1/n)=lim_(n-gtoo) n^(1/n) ((7/8)^n)^(1/n)`


                               `=lim_(n-gtoo) n^(1/n) (7/8)^(n*1/n)`


                               `=lim_(n-gtoo) n^(1/n) (7/8)^(n/n)`


                               `=lim_(n-gtoo) n^(1/n) (7/8)^1`


                               `=lim_(n-gtoo) 7/8n^(1/n)`


Evaluate the limit.


`lim_(n-gtoo) 7/8n^(1/n) =7/8 lim_(n-gtoo) n^(1/n) `           


                   ` =7/8 *1`


                   ` =7/8 or 0.875`


The limit value `L =7/8 or 0.875` satisfies the condition: L<1 since `7/8lt1 or 0.875lt1.`


Conclusion: The series `sum_(n=1)^oo n(7/8)^n` is  absolutely convergent.

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