Use the limit comparison test to determine whether ${\displaystyle \underset{n=6}{\overset{}{}}}{a}_n={\displaystyle \underset{n=6}{\overset{}{}}}\frac{4n^3-8n^2+6}{8+9n^4}$ converges or diverges.

$($a$)$ Choose a series ${\displaystyle \underset{n=6}{\overset{}{}}}{b}_n$ with terms of the form $b_n=\frac{1}{n^p}$ and apply the limit comparison test. Write your answer as a fully simplified fraction.

For $n6,$

$\underset{n}{lim}\frac{a_n}{b_n}=\underset{n}{lim}$

$($b$)$ Evaluate the limit in the previous part. Enter $$ as infinity and $--$ as $-$infinity, If the limit does not exist, enter DNE.

$\underset{n}{lim}\frac{a_n}{b_n}=\frac{4}{9}$

$($c$)$ By the limit comparison test, does the series converge, diverge, or is the test inconclusive?