Series of functions

A series of functions on a set $D$ is an expression

\sum_{n = 1}^{\infty} f_{n} (x), x \in D,

understood via its partial sums

S_{N} (x) = \sum_{n = 1}^{N} f_{n} (x) .

The series converges (to $S$ ) in some mode if $S_{N} \to S$ in that mode.

Modes of convergence

Pointwise convergence

$S_{N} \to S$ pointwise on $D$ if for every $x \in D$ we have $S_{N} (x) \to S (x)$ .

Uniform convergence

$S_{N} \to S$ uniformly on $D$ means the convergence is uniform in $x$ (so the same $N_{0}$ works for all $x \in D$ ).

This is the main mode of convergence used to justify swapping limits with operations.
See: uniform convergence (definition, Cauchy criterion, M-test, and standard consequences).

Canonical examples

Power series

A power series

\sum_{n = 0}^{\infty} a_{n} (z - z_{0})^{n}

converges uniformly on compact subsets of its disk of convergence $D (z_{0}, R)$ . This is the key analytic reason term-by-term differentiation/integration works inside the radius.

Radius of convergence

The radius of convergence $R \in [0, \infty]$ characterizes the disk $| z - z_{0} | < R$ where the series converges absolutely. Within this disk, the convergence is uniform on compact sets. Outside, for $| z - z_{0} | > R$ , the series diverges.

The radius $R$ is determined by the following criteria:

Cauchy-Hadamard Formula: This is the general definition: $\frac{1}{R} = \underset{n \to \infty}{lim sup} \sqrt[n]{| a_{n} |}$
Ratio Test (D'Alembert criterion): If the following limit exists (or is infinite): $R = lim_{n \to \infty} | \frac{a_{n}}{a_{n + 1}} |$

On the boundary $| z - z_{0} | = R$ , the series may converge or diverge; this behavior must be studied case-by-case.

See: analytic function and Taylor's theorem

Laurent series

A Laurent series in an annulus $r_{1} < | z - z_{0} | < r_{2}$ converges absolutely in the annulus and uniformly on compact subannuli.
See: Laurent series.

Fourier series

Fourier series are often discussed in norms like $L^{2}$ :

{‖ S_{N} - f ‖}_{L^{2}} \to 0,

which is weaker than uniform convergence. As a result, pointwise/uniform statements and term-by-term manipulations generally need extra hypotheses.
See: Fourier series expansion and isomorphism between l_2 and L_2.

Stone–Weierstrass theorem (uniform approximation of continuous functions)