Problem $2.$

$(1$ point$)$ Use the ratio test to determine whether ${\displaystyle \underset{n=29}{\overset{}{}}}\frac{n(-3{)}^n}{n!}$ converges or diverges.

$($a$)$ Find the ratio of successive terms. Write your answer as a fully simplified fraction. For $n29,$

$\underset{n}{lim}|\frac{a_{n+1}}{a_n}|=\underset{n}{lim}$

$($b$)$ Evaluate the limit in the previous part. Enter $$ as infinity and $-$ as $-$infinity. If the limit does not exist, enter DNE. $\underset{n}{lim}|\frac{a_{n+1}}{a_n}|=$

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

Note: You can earn $40\%$ partial credit for $2-3$ correct answers.