Answer
The radius of convergence: 4
The interval of convergence: $(-5,3)$
Work Step by Step
Let $a_n=\frac{n(x+1)^n}{4^n}$.
It can be checked that $\frac{a_{n+1}}{a_n}=\frac{(n+1)(x+1)}{4n}$.
\We know that $\lim \frac{n+1}{4n}=\frac{1}{4}$.
Then,
$\lim |\frac{a_{n+1}}{a_n}|=|\frac{x+1}{4}|$
Find the interval of convergence:
$|\frac{x+1}{4}|<1$
$|x+1|<4$
$(-5,3)$
