Answer
converges to $\dfrac{6}{3-x}$ for $ -1 \lt \dfrac{x-1}{2} \lt 1$ or, $ -1 \lt x \lt 3$
Work Step by Step
Formula to calculate the sum of a geometric series is:
$S=\dfrac{a}{1-r}$;
Here, $a=3$ and common ratio $r =\dfrac{x-1}{2}$
$S=\dfrac{a}{1-r}=\dfrac{3}{1-(\dfrac{x-1}{2})}$
or, $S=\dfrac{6}{3-x}$
Hence, the series converges to $\dfrac{6}{3-x}$ for $ -1 \lt \dfrac{x-1}{2} \lt 1$
or, $ -1 \lt x \lt 3$