Answer
$$R=2$$
and
Interval of convergence: $$[-2,2)$$
Work Step by Step
$$|\frac{a_{n+1}}{a_{n}}|=|\dfrac{n+1}{n}\dfrac{x^{n+1}}{x^n}\dfrac{2^n}{2^{n+1}}\dfrac{n^2+1}{n^2+2n+2}|$$
Take limits on both sides, we have $$\lim\limits_{n \to \infty}|\frac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{n+1}{n}\dfrac{x^{n+1}}{x^n}\dfrac{2^n}{2^{n+1}}\dfrac{n^2+1}{n^2+2n+2}|=\dfrac{|x|}{2}$$
Hence, $$R=2$$
and
Interval of convergence: $[-2,2)$
