Find the radius of convergence $R$ and the interval of convergence of the power series $$\sum\limits_{k\,=\,0}^{\infty }\dfrac{x^{k}}{(k+2) ^{2k}}$$

Solution We use the Root Test. Then $$\lim\limits_{n\rightarrow \infty }\sqrt[n]{\left|\dfrac{x^{n}}{(n+2) ^{2n}}\right|}=\lim\limits_{n \rightarrow \infty }\dfrac{\vert x\vert }{(n+2) ^{2}} =\vert x\vert \lim\limits_{n\rightarrow \infty }\dfrac{1}{(n+2) ^{2}}=0$$

The series converges absolutely for all $x$. The radius of convergence is $R=\infty$, and the interval of convergence is $(-\infty ,\infty)$.