Pointwise summable majorants give pointwise absolute convergence.Suggest a slogan
Let \(S\) be a set and let \(f_n\colon S\to\CC\) for \(n\geq1\). Suppose that for each \(x\in S\) there are constants \(M_n(x)\geq0\) with \(\abs{f_n(x)} \leq M_n(x)\) for all \(n\) and \(\sum_{n} M_n(x) < \infty\). Then the series \(f(x) \coloneqq \sum_{n} f_n(x)\) converges absolutely at every \(x\in S\).
The constants \(M_n(x)\) may depend on \(x\), and the conclusion is only pointwise convergence. When one sequence \(M_n\) serves every \(x\in S\), the Weierstrass $M$-test gives uniform convergence on \(S\).