The Weierstrass M-Test

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $D$ be a set, let $S\subseteq D$, let $(g_{k})_{k\in\mathbb{N}}$ be a sequence in the set $\mathbb{R}^{D}$ of maps from $D$ to $\mathbb{R}$ of Sets and Maps: Ordinary Notation (v), and let $(M_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}$ such that the series $\sum_{k=1}^{\infty}M_{k}$ converges and

$$|g_{k}(x)|\le M_{k}\qquad\text{for every }k\in\mathbb{N}\text{ and every }x\in S .$$

For $n\in\mathbb{N}$ let $s_{n}:D\to\mathbb{R}$ be the map $x\mapsto\sum_{k=1}^{n}g_{k}(x)$ (Sequences of Real-Valued Maps: the Map of Pointwise Limits on a Subset, and the Sequence of Pointwise Partial Sums (ii)). For $x\in S$, $\sum_{k=1}^{\infty}g_{k}(x)$ is the series of the sequence $(g_{k}(x))_{k\in\mathbb{N}}$ of The Sequences k ↦ f_k(p) of a Sequence in ℝ^I, and the Map of Their Limits (i), and for $n\in\mathbb{N}$, $\sum_{k=1}^{\infty}M_{n+k}$ is the series of the sequence $(M_{n+k})_{k\in\mathbb{N}}$. Convergence and sums of series are as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii), absolute convergence as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (iii), and uniform convergence as in Pointwise and Uniform Convergence of a Sequence of Real-Valued Maps (ii).

(i) For every $x\in S$ the series $\sum_{k=1}^{\infty}g_{k}(x)$ converges absolutely and converges, and there is exactly one map $s:S\to\mathbb{R}$ with $s(x)=\sum_{k=1}^{\infty}g_{k}(x)$ for every $x\in S$.

(ii) For every $n\in\mathbb{N}$ the series $\sum_{k=1}^{\infty}M_{n+k}$ converges, and the map $s$ of (i) satisfies

$$|s(x)-s_{n}(x)|\le\sum_{k=1}^{\infty}M_{n+k}\qquad\text{for every }x\in S .$$

Moreover, the sequence $\big(\sum_{k=1}^{\infty}M_{n+k}\big)_{n\in\mathbb{N}}$ converges to $0$; and if $M_{k}\ge0$ for every $k\in\mathbb{N}$, then $\sum_{k=1}^{\infty}M_{n+k}\ge0$ for every $n\in\mathbb{N}$.

(iii) The sequence $(s_{n})_{n\in\mathbb{N}}$ converges uniformly on $S$ to the map $s$ of (i), and also to every map $f:E\to\mathbb{R}$ on a set $E$ with $S\subseteq E$ such that $f(x)=\sum_{k=1}^{\infty}g_{k}(x)$ for every $x\in S$.