Finding the Interval of Convergence of a Power Series

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 \((-\infty ,\infty)\).