Sum

Sum is the result of adding a sequence of numbers. In mathematics, a finite sum is written with the summation symbol Σ: Σ_{i=m}^n a_i. The value is the partial sum S_n = Σ_{i=1}^n a_i. Summation is linear: Σ(a_i + b_i) = Σ a_i + Σ b_i and Σ c a_i = c Σ a_i. The order of addition does not affect a finite sum due to the commutativity of addition.

Finite sums: Arithmetic series involve consecutive terms from an arithmetic progression. The sum of the first n terms is S_n = n/2 (a_1 + a_n). For example, the sum of the integers from 1 to n is n(n+1)/2. Geometric series have a common ratio r. The sum of the first n terms is S_n = a_1 (1 - r^n)/(1 - r) for r ≠ 1. If |r| < 1, the infinite geometric series Σ a_1 r^{n-1} converges to a_1/(1 - r).

Infinite sums: A series is the limit of its partial sums as n grows. If this limit exists and is finite, the series converges; otherwise it diverges. Some series telescope, where successive terms cancel and yield a simple limit.

Applications and contexts: Sums are used to compute totals, averages, probabilities, and in analysis and physics. The concept extends to vectors and functions, and in many cases can be related to continuous sums through the definite integral as a limiting process.