Uniform convergence of a function series forces its terms to vanish uniformly.Suggest a slogan
Let \(S\) be a set and let \(f_n\colon S\to\CC\) for \(n\geq1\).
If \(\sum_{n\geq1}f_n(x)\) converges for every \(x\in S\), then \(f_n\to0\) pointwise on \(S\).
If \(\sum_{n\geq1}f_n\) converges uniformly on \(S\), then \(f_n\to0\) uniformly on \(S\).
Pointwise convergence of the series does not give uniform convergence of the terms: on \(S\coloneqq[0,1)\), \(\sum_{n\geq0}x^n\) converges for every \(x\in S\), but \(\sup_{x\in S}\abs{x^n}=1\) for all \(n\).