$(4$ pts$)10)$ Briefly explain what is wrong with the following attempt to find the interval of convergence for the power series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n!(x+2{)}^n}{n^2}$

Use the Ratio Test:

$\underset{n}{lim}|\frac{a_{n+1}}{a_n}|<mn>1</mn>$

$\underset{n}{lim}|\frac{(n+1)!(x+2{)}^{n+1}}{(n+1{)}^2}*\frac{n^2}{n!(x+2{)}^n}|<mn>1</mn>$

$\underset{n}{lim}|\frac{(n+1)(x+2)n^2}{(n+1{)}^2}|<mn>1</mn>$

$\underset{n}{lim}|\frac{(x+2)n^2}{(n+1)}|<mn>1</mn>$

$|x+2|\underset{n}{lim}\frac{n^2}{(n+1)}<mn>1</mn>$

$|x+2|*<mn>1</mn>$

$<mn>1</mn>$

The limit is never less than $1$ and the power series never converges.