Calculus of a Single Variable, Chapter 9, 9.6, Section 9.6, Problem 18

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.