section_path: Generalizations/Summations over arbitrary index sets
section_name: Summations over arbitrary index sets
paragraph_text: Summations over arbitrary index sets
  
  Definitions may be given for sums over an arbitrary index set I. {\displaystyle I.} There are two main differences with the usual notion of series: first, there is no specific order given on the set I {\displaystyle I}; second, this set I {\displaystyle I} may be uncountable. The notion of convergence needs to be strengthened, because the concept of conditional convergence depends on the ordering of the index set.
  
  If a: I ↦ G {\displaystyle a:I\mapsto G} is a function from an index set I {\displaystyle I} to a set G, {\displaystyle G,} then the "series" associated to a {\displaystyle a} is the formal sum of the elements a (x) ∈ G {\displaystyle a(x)\in G} over the index elements x ∈ I {\displaystyle x\in I} denoted by the
  
  ∑ x ∈ I a (x). {\displaystyle \sum _{x\in I}a(x).}
  
  When the index set is the natural numbers I = N, {\displaystyle I=\mathbb {N},} the function a: N ↦ G {\displaystyle a:\mathbb {N} \mapsto G} is a sequence denoted by a (n) = a n. {\displaystyle a(n)=a_{n}.} A series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N {\textstyle \sum _{n\in \mathbb {N} }} as ∑ n = 0 ∞ {\textstyle \sum _{n=0}^{\infty }} in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers
  
  ∑ n = 0 ∞ a n = a 0 + a 1 + a 2 + ⋯. {\displaystyle \sum _{n=0}^{\infty }a_{n}=a_{0}+a_{1}+a_{2}+\cdots.}