Tutorial Exercise

Use the Ratio Test to determine whether the series is convergent or divergent.

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n!}{{91}^n}$

Step $1$

Recall that for the series ${\displaystyle \underset{n=0}{\overset{}{}}}{a}_n$ the Ratio Test looks at the limit $\underset{n}{lim}|\frac{a_{n+1}}{a_n}|.$The value of this limit can determine if the series absolutely converges, diverges, or the test can be inconclusive. We are given the series ${\displaystyle \underset{n=0}{\overset{}{}}}\frac{n!}{{91}^n}.$ Identify the following terms.

$a_n=$

$a_{n+1}=$