Answer
The radius of convergence of the series is $5$.
Work Step by Step
Here, we have $a_n=\dfrac{(-1)^n(x)^n}{5^n}$
Root Test states that when $\Sigma a_n$ is an infinite series with positive terms and, then $r=\lim\limits_{n \to \infty}\sqrt[n] a_n$
a) When $0 \leq r \lt 1$, the series converges. (b) When $r \gt 1$, or, $\infty$, so the series diverges. (c) When $r=1$, the ratio test is inconclusive.
Now, $r=\lim\limits_{n \to \infty}\sqrt[n] {\dfrac{x^n}{5^n}}\\=\lim\limits_{n \to \infty} |{\dfrac{x}{5}}|\\=|{\dfrac{x}{5}}|$
Therefore, the series converges for $|{\dfrac{x}{5}}| \lt 1 \implies |x| \lt 5$ by the root test. This implies that the radius of convergence of the series is $5$.
