prove that the following series defines differentiable function on $(0,\infty)$
$$\sum_{n=0}^\infty x^2e^{-nx}$$ I tried to show:
1) $f_n(x)=x^2e^{-nx}$ is differentiable -simple.
2) $\sum_{n=0}^\infty x^2e^{-nx}$ convergent for one point- also simple (for instance $x=1$).
3)$\sum_{n=0}^\infty (f_n(x))'$ uniformly convergent.
I got stuck on 3). I'm was thinking about using M-test, but failed to find a series which is bigger than then this. I tried to find maximum to $(f_n(x))'$ and failed as well.