Series of functions

Definition

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 0$. The series of functions $\sum_{n=0}^{\infty} f_n$ is the sequence of partial sums $s_N = \sum_{n=0}^{N} f_n$. The series converges pointwise, or uniformly, on $S$ if the sequence $(s_N)$ converges pointwise, or uniformly, on $S$. The limit is the sum $\sum_{n=0}^{\infty} f_n(x)$.

Remarks

Power series and Fourier series are series of functions. The Weierstrass M-test is the main tool to prove uniform convergence. By the uniform limit theorem, a uniformly convergent series of continuous functions has a continuous sum.

Explanations

L1 human

A series of functions adds infinitely many functions, point by point. At each $x$ it is an ordinary series of numbers.