Answer
The radius of convergence is equal to: $\dfrac{5}{2}$
Work Step by Step
Apply the Ratio Test to the series as follows:
$$\lim\limits_{n \to \infty} |\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} |\dfrac{(2n+3) (x-1)}{5n+4}| \\=|x-1| \times \lim\limits_{n \to \infty} (\dfrac{(2n+3)}{5n+4}) \\=\dfrac{2}{5}|x-1|$$
But the series is absolutely convergent for $\dfrac{2}{5}|x-1| \lt 1 \implies |x-1| \lt \dfrac{5}{2}$
Thus, the radius of convergence is equal to: $\dfrac{5}{2}$
