Find the Taylor series centered at $c=-1.$

$f(x)=e^{4x}$

Identify the correct expansion.

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{x^n{e}^{-4}}{n!}$

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{4^n{e}^{-4}}{n!}(x1{)}^n$

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{4^n}{n!}(x1{)}^n$

${\displaystyle \underset{n=0}{\overset{}{}}}\frac{4^{n-4}}{n!}(x1{)}^n$

Find the interval on which the expansion is valid.

$($Give your answer as an interval in the form $(**).$ Use the symbol prop for infinity, $$ for combining intervals, and an appropriate type of parenthesis $"(,),7,$ depending on whether the interval is open or closed. Enter $\frac{?}{\text{b}\text{a}\text{r}}(8)$ if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.$)$