term-by-term differentiation

Differentiating a series term by term — (sum f_n)' = sum f_n' — looks like the natural sibling of term-by-term integration, but it is far more dangerous. Differentiation amplifies wiggles, so uniform convergence of the f_n themselves is NOT enough. The convergence must be imposed on the DERIVATIVES, which is a strictly stronger demand.

Theorem: suppose each f_n is differentiable on an interval [a, b], the derivatives f_n' converge uniformly on [a, b], and at just one point x_0 the values f_n(x_0) converge. Then f_n converges uniformly on [a, b] to a differentiable function f, and f'(x) = lim f_n'(x) for every x. In series form, sum f_n converges to a differentiable sum whose derivative is sum f_n', provided sum f_n' converges uniformly and the series converges at one point.

The reason for the lopsided hypotheses: integration smooths, differentiation roughens. A classic warning example is f_n(x) = sin(nx)/sqrt(n): these converge uniformly to 0 (sup = 1/sqrt(n) -> 0), yet f_n'(x) = sqrt(n) cos(nx) does not converge at all. So even uniform convergence of the functions tells you nothing about the limit of their derivatives — you genuinely need to control the derivatives directly.