I I'm interested in, if it is true:
If series $\sum f_k$ is uniformly convergence on every interval $[n; n+10]$, where $n \in \mathbb{Z}$, then $\sum f_k$ is uniformly convergence on $\mathbb{R}$.
Infinitely many intervals doesn't looks so good, so I'm not sure, if I can do task like that.
I have to check if $\sum_{k=0}^{\infty}\frac{1}{2+x^2 - k^2}$ is uniformly convergence on $\mathbb{R}$, and my idea was to split it to intervals. However, I don't know, if I can do something like that. Intuition tells me that's absolutely wrong, but I heard something else and I want to know. I will appreciate answer.
On the first question the answer is negative. For example, $f_k(x) = x^2/k$ converges uniformly to $0$ on every bounded interval, but does not converge uniformly to $0$ on $\mathbb R.$
For your series, here's a useful fact: If $\sum_k f_k$ converges uniformly on a set $E,$ then $f_k \to 0$ uniformly on $E.$ Is that true here?