Perform the ratio test.
$$ \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| \lt 1 $$
$$ \lim_{n\to\infty}\frac{(x-3)^{n+1}}{(n+1)\cdot 2^{n+1}}\cdot \frac{n\cdot 2^n}{(x-3)^n} $$
$$ =\lim_{n\to\infty}\frac{n(x-3)}{2(n+1)} $$
$$ \left| \frac{x-3}{2} \right|\lt 1 $$
$$ \left|x-3\right| \lt 2 $$
$$ 1\lt x \lt 5 $$
\(x=1\)
$$ \sum_{n=1}^\infty \frac{(1-3)^n}{n\cdot 2^n} $$
$$ \sum_{n=1}^\infty \frac{(-2)^n}{n\cdot 2^n} $$
$$ = \sum_{n=1}^\infty \frac{(-1)^n}{n} $$
Which converges conditionally through the alternating series test.
\(x=5\)
$$ \sum_{n=1}^\infty \frac{(5-3)^n}{n\cdot 2^n} $$
$$ \sum_{n=1}^\infty \frac{(2)^n}{n\cdot 2^n} $$
$$ = \sum_{n=1}^\infty \frac{1}{n} $$
Which is a \(p\)-series that diverges.