Answer
The series has a radius of convergence $5$ and interval $[-5,5]$.
Work Step by Step
Root test:$\lim\limits_{n \to \infty}\sqrt[n] {\frac{|x^{n}|}{n^{2}5^{n}}}=\frac{|x|}{5}\lt 1$
$|x|\lt 5$
$-5\lt x\lt 5$
Radius of convergence = 5
For $x=5$
$\Sigma_{1}^{\infty}(-1)^{n}\frac{5^{n}}{n^{2}5^{n}}=\Sigma_{1}^{\infty}(-1)^{n}\frac{1}{n^{2}}(\frac{5}{5})^{n}$
$=\Sigma_{1}^{\infty}(-1)^{n}\frac{1}{n^{2}}$
Thus, the series has a radius of convergence $5$ and interval $[-5,5]$.
