Is the $\sum_{n=1}^\infty \frac{nx}{1+n^3x^2} $ uniformly convergent on $(0,\infty)$?
Each individual term attains a max of $\frac{1}{2n^{1/2}} $ at $x=\frac{1}{n^{1.5}}$.
Notice that the maximizers move to the left and maximums shrink as $n$ increases. So, one cannot put a lower bound of $\sum \frac{1}{2n^{0.5}}$ in a crude way. But I believe with some more algebra we might still bound from below by the latter sum which diverges. And that is what I am asking!