Answer
The series has a radius of convergence $4$ and interval $[-6,2)$.
Work Step by Step
Root test:$R=\lim\limits_{n \to \infty}\sqrt[n] {{n4^{n}}}=\lim\limits_{n \to \infty}4\sqrt[n] {n}=4.1=4$
Radius of convergence $=4$
$-4\lt x+2\lt 4$
$-4-2\lt x\lt 4-2$
$-6\lt x\lt 4$
When $x=-6$
Series $\Sigma_{0}^{\infty}\frac{(-1)^{n}}{n}$converges by AST.
When $x=2$
Series $\Sigma_{0}^{\infty}\frac{1}{n}$, which diverges because it is a p-series with $p=1$
Thus, the series has radius of convergence $4$ and interval $[-6,2)$.
