The problem states as follows:
Test whether the following sequence converges uniformly on $(0,\infty)$
$$ f_n(x)=\sum_{k=0}^n \arctan\frac{k^2x}{n^3} $$
Firstly I don't know how to find this sum or if it is even possible. I have tried finding a pattern in first few terms and with intention to find a formula which I would later prove inductively with no luck.
I have however managed to prove pointwise convergence by finding an upper bound ($\frac{x}{3}$) and noticing that the sequence is increasing I deduced that it must converge.
In this situation I would usually use Cauchy's test for uniform convergence, but I had no luck with it.
Hint: If the sequence $f_n(x)$ converges uniformly on $(0,\infty)$, then
$$f_n(x)-f_{n-1}(x)=\arctan \left (\frac{n^2x}{n^3}\right ) \to 0\,\,\text{uniformly on}\,(0,\infty).$$