summations

Summations, or summation notation, is the operation of combining the terms of a sequence by addition. The standard notation uses the sigma symbol ∑, with an index i that runs over a specified range. A finite sum is written as ∑_{i=m}^n a_i, summing terms a_i from i=m to i=n. An infinite series is written as ∑_{i=m}^∞ a_i and is defined as the limit of the finite sums S_n = ∑_{i=m}^n a_i as n → ∞, when this limit exists.

Common examples include arithmetic series, where a_i forms an arithmetic progression, and geometric series, where a_i = a r^i. The finite sums have closed forms: ∑_{i=1}^n i = n(n+1)/2 for the first n integers; for a geometric series with ratio |r|<1, ∑_{i=0}^∞ a r^i = a/(1−r).

Convergence and divergence distinguish infinite series that approach a finite limit from those that do not. Important properties include linearity: ∑ (c_i + d_i) = ∑ c_i + ∑ d_i and ∑ c a_i = c ∑ a_i, plus reindexing and grouping rules that allow rearranging or grouping terms under certain conditions.

Applications appear throughout mathematics and applied disciplines, including probability, numerical analysis, and physics, wherever cumulative totals or expansions are needed.