A pointwise-convergent series is uniformly convergent exactly when its tails vanish uniformly.Suggest a slogan
Let \(X\) be a set and let \(f_n\colon X\to\CC\) for \(n\ge1\) be functions such that \(\sum_{n=1}^\infty f_n(x)\) converges for every \(x\in X\). Then \(\sum_{n=1}^\infty f_n\) converges uniformly on \(X\) if and only if \[ \lim_{n\to \infty} \sup_{x\in X}\abs{ \sum_{k\geq n} f_k(x) } = 0. \]
Let \(S(x)\coloneqq\sum_{k\ge1}f_k(x)\) and \(S_n(x)\coloneqq\sum_{k=1}^nf_k(x)\). Then \(\sum_{k\ge n}f_k(x)=S(x)-S_{n-1}(x)\), so the displayed condition says exactly that \(\sup_{x\in X}\abs{S(x)-S_{n-1}(x)}\to0\), which is uniform convergence of the partial sums \(S_n\) to \(S\).