Answer
Radius of convergence is $8$
Work Step by Step
We need to apply the Ratio Test to the series.
$\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_n}|=\lim\limits_{n \to \infty} \dfrac{\dfrac{(n+1)a_nx}{4(2n+1}}{a_n}$
or, $=|x| \lim\limits_{n \to \infty} \dfrac{n+1}{8n+4}$
or, $=|x| \lim\limits_{n \to \infty} \dfrac{1+1/n}{8+4/n}$
or, $=\dfrac{1}{8}|x|$
Now, $\dfrac{1}{8}|x| \lt 1 \implies |x| \lt 8$
Thus, the radius of convergence is $8$.
