$\lim \limits_{k\to\infty}\Big|\frac{(-4)^{k+1}(x-3)^{k+1}}{k+1}\cdot\frac{k}{(-4)^k(x-3)^k}\Big|<1$

$|x-3|\lim \limits_{k\to\infty}\Big|\frac{4k}{k+1}\Big|<1$

$2\frac{3}{4}<x<3\frac{1}{4}$

Now test the convergence of the series at each endpoint of the interval.

$\frac{(k+1)!}{(k+2)!}=\frac{1}{k+2}$

$=\displaystyle \sum_{k=0}^\infty \frac{(3x)^k}{k}$

After applying the Ratio Test, you should get:

$3|x|<1$

Now test the convergence of the series using the endpoints of the interval.