Definition:Uniform Convergence/Infinite Series

Definition

Let $S \subseteq \R$.

Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.

Let $\sequence {s_n}$ be sequence of real functions $S \to \R$ with:

$\ds \map {s_n} x = \sum_{k \mathop = 1}^n \map {f_n} x$

for each $n \in \N$ and $x \in S$.

We say that:

$\ds \sum_{n \mathop = 1}^\infty f_n$

converges uniformly to a real function $f: S \to \R$ on $S$ if and only if $\sequence {s_n}$ converges uniformly to $f$ on $S$.

Sources

• 1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 9.6$: Uniform Convergence of Infinite Series of Functions: Definition $9.4$