## Thomas' Calculus 13th Edition

$8$
Let us appy the Ratio Test to the given series. In order to find the radius of convergence, we have: $\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_n}|=|\lim\limits_{n \to \infty} \dfrac{\dfrac{(n+1)a_nx}{4(2n+1)}}{a_n}| \\=|x| \lim\limits_{n \to \infty} \dfrac{1+\dfrac{1}{n}}{8+(4/n)} \\=\dfrac{1}{8}|x|$ Now, $\dfrac{1}{8}|x| \lt 1$ and $|x| \lt 8$ Thus, the radius of convergence is $8$