$\sum_{n=1}^{\infty}\dfrac{1}{n(1+(x-n)^2)}$
The question has a few parts and the one im struggling with is checking for uniform convergence on $[0,\infty)$.
I've managed to show that this series converges for any $x\in[0,\infty)$ and that the seriesconverges uniformly on every $[a,b]\subset[0,\infty)$. My intuition tells me that the series should be uniformly continuous, because even though it's terms are sometimes close to $\dfrac{1}{n}$ when $n$ is close to $x$, almost all terms are proportional to $\dfrac{1}{n^3}$.
I feel like I'm overlooking something obvious here. I would appreciate any hints.