So, here is the series: $$\Sigma_{k=1}^{\infty} \arctan(x^6ne^{-nx^2})$$ I try to find sup series by deriving the main term: $$\frac{1}{1 + (x^6ne^{-nx^2})^2} \cdot n(6x^5e^{-nx^2} - 2nx^7e^{-nx^2})$$
but I think I did something incorrectly because I can't express $x$ out of this. Function $\arctan x$ is not bounded by a constant so I cannot get the sup series by bounding the function, maybe I can use as a bounding series something like $\sum^{\infty}_{k=1} \arctan(x^6 n )$, but still I don't know how to proceed.
Let $[a,b]\subset\mathbb{R}$ and $x\in[a,b]$ then $$ \forall x\in\mathbb{R},\left|\arctan\left(x^6ne^{-nx^2}\right)\right|\leqslant x^6n e^{-nx^2}\leqslant b^6ne^{-na^2} $$ and the series $\sum b^6ne^{-na^2}$ converges, thus the series converges uniformly on the segments of $\mathbb{R}$.