Use the Ratio Test or the Root Test to determine whether the following series converges absolutely or diverges.

${\displaystyle \underset{k=1}{\overset{}{}}}(-1{)}^{k+1}(\frac{7k^4+k}{5k^4+k+1}{)}^k$

Select the correct choice below and fill in the answer box within your choice.

$($Simplify your answer.$)$

A$.$ The series diverges by the Root Test because $=$

B$.$ The Root Test is inconclusive because $=$

C$.$ The series converges absolutely by the Root Test because $=$

D$.$ The Ratio Test is inconclusive because $r=$

E$.$ The series converges absolutely by the Ratio Test because $r=$

F$.$ The series diverges by the Ratio Test because $r=$