Pointwise convergence of $\sum_{n=0}^\infty \frac{1}{2^n\sqrt{1+nx}}$

Question

Given this series : $$ \sum_{n=0}^\infty \frac{1}{2^n*\sqrt{1+nx}} $$ I have to prove for which $x \geq 0$ the series converges pointwise. if $x=0$ the series is :

$$\sum_{n=0}^\infty \Big(\frac{1}{2}\Big)^n$$

and this series converges.

If I assume that $x\neq 0$ I don't find the convergence. How can I proceed? Do I have to consider a majorant series? Thanks in advance for any help.