Answer
Radius of convergence is: $\dfrac{5}{2}$
Work Step by Step
We need to apply the Ratio Test to the series.
$\lim\limits_{n \to \infty} |\dfrac{u_{n+1}}{u_n}|=\lim\limits_{n \to \infty} |\dfrac{(2n+3) (x-1)}{5n+4}|$
or, $=|x-1| \lim\limits_{n \to \infty} (\dfrac{(2n+3)}{5n+4})$
or, $=|x-1| \lim\limits_{n \to \infty} (\dfrac{(2+3/n)}{5+4/n})$
So, $=\dfrac{2}{5}|x-1|$
The series converges absolutely for $\dfrac{2}{5}|x-1| \lt 1$ or, $|x-1| \lt \dfrac{5}{2}$
So, the radius of convergence is: $\dfrac{5}{2}$
