As the title indicates I am slightly unsure about a setup used for convergence of a series.
I was wondering if there is/what the difference is between showing that a series is uniformly convergent on $\Bbb R$ vs. uniformly convergent in any interval $[-K,K]$, for fixed $K$ where $0<K<\infty$.
If I want to show that the sum function is continuous in any $x$ in the real numbers I would just choose $K=\text{numerical($x$)}$ and obtain continuity on $[-x,x]$, right?
I will assume that your series is made up of terms which are continuous. The sum function being continuous is only a necessary condition for uniform convergence. It is not sufficient. The series $\sum \frac x {n^{2}}$ is uniformly convergent on $[-K,K]$ for any $K >0$ (by M-test) but it is not uniformly convergent on $\mathbb R$ because the general term $\frac x {n^{2}}$ does not tend to $0$ uniformly on $\mathbb R$. The sum is $\frac {\pi^{2}} 6 x$ which is continuous on the whole line but still the series is not uniformly convergent.