---- The following concepts are related but distinct. Neither subsumes the other. > [!definition] Definition. ([[series|sum over set]]) > Let $S$ be a set, and let $f:S \to [0, \infty]$ be a function. The **sum of $f$ over $S$** is defined by the formula $\sum_{s \in S}f(s)=\sup \left\{ \sum_{n=1}^{N} f(s_{n}): \{ s_{1},\dots,s_{N} \} \text{ is a finite subset of }S \right\}.$ It can equal $\infty$. ^definition > [!definition] Definition.([[series]]) > Let $(a_{n})$ be a [[sequence]] in (say) a [[norm|normed space]]. The **series** $\sum_{n=1}^{\infty}a_{n}$ is defined as the limit of partial sums $\sum_{n=1}^{\infty} a_{n}:= \lim_{N \to \infty} S_{N}, \text{ where } S_{N}=\sum_{n=1}^{N} a_{n}$ > if that limit exists. In this case, if $\sum_{n=1}^{\infty} \|a_{n}\|<\infty$ then we say $\sum_{n=1}^{\infty} a_{n}$ **converges absolutely**, and if not then we say it **converges conditionally**. ^definition > [!specialization] Specializing. (When the two definitions intersect) > When $S=\mathbb{N}$ in the first definition, $f:\mathbb{N} \to [0, \infty)$ is a [[sequence]] of nonnegative numbers $f(1)=a_{1}, f(2)=a_{2},\dots$ and (by the [[monotone convergence theorem for sequences]]) the sum of $f$ over $S$ equals the series $\sum_{n=1}^{\infty}a_{n}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```