Sequences and Series of Functions

Abstract

• We say that a sequence of functions {f _n : D → ℝ} defined on a subset D ⊆ ℝ converges pointwise on D if for each x ∈ D the sequence of numbers {f _n(x)} converge. If {f _n} converges pointwise on D, then we define f : D → ℝ with $f\left( x \right)\, = \,\mathop {\lim }\limits_{n \to \infty } \,f_n \left( x \right)$ for each x ∈ D. We denote this symbolically by f _n → f on D.

Where is it proved that one obtains the derivative of an infinite series by taking derivative of each term?

Niels Henrik Abel (1802–1829)