Finite sum - Knowlpedia

For elements $a_1,\ldots,a_n$ of an additive abelian group, the finite sum is defined recursively by

\sum_{j=1}^{0}a_j=0,\qquad \sum_{j=1}^{n+1}a_j=\left(\sum_{j=1}^{n}a_j\right)+a_{n+1}.

Associativity makes parenthesization immaterial, and commutativity permits arbitrary reordering. Hence for a family $(a_i)_{i\in I}$ indexed by a finite set, the notation $\sum_{i\in I}a_i$ does not require an ordering of $I$ .

Grouping

Splitting a finite index set into disjoint subsets splits the sum into the sum over those subsets. No convergence condition is required for a finite sum. An infinite series instead requires a definition of convergence of its partial sums.