## Calculus 8th Edition

series has radius of convergence $4$ and interval $[-6,2)$
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 $[-6,2)$