Uniform convergence of the derivative series plus convergence at one point permits term-by-term differentiation.Suggest a slogan
Let \(a<b\), let \(A\coloneqq[a,b]\), and let \(f_n\colon A\to\RR\) for \(n\geq1\) be differentiable on \(A\). Suppose that \(\sum_{n=1}^\infty f_n'\) converges uniformly on \(A\) to a function \(g\colon A\to\RR\), and that \(\sum_{n=1}^\infty f_n(x_0)\) converges for some \(x_0\in A\). Then \(\sum_{n=1}^\infty f_n\) converges uniformly on \(A\) to a function \(f\colon A\to\RR\), \(f\) is differentiable on \(A\), and \(f'=g\) on \(A\) [@Abb16].