Sums

In mathematics, a sum is the result of adding two or more numbers or expressions. The operation of addition is commutative and associative, and 0 acts as the additive identity. The sum of a finite sequence a1, a2, ..., an is written as a1 + a2 + ... + an, and is often denoted by the sigma notation sum_{i=1}^n a_i, where i is a summation index from a lower bound to an upper bound.

Finite sums can be analyzed with formulas. The sum of the first n natural numbers is n(n+1)/2. The sum of an arithmetic series with first term a1 and last term an and n terms is n/2 (a1 + an). For a geometric series with first term a and common ratio r, the finite sum is a (1 - r^n)/(1 - r) when r ≠ 1.

Infinite sums, or series, extend to infinity. The infinite geometric series sum_{i=0}^{∞} ar^i converges to a/(1 - r) if |r| < 1; otherwise it diverges. Some infinite sums converge conditionally or absolutely, depending on the terms. Techniques such as telescoping sums simplify certain series by cancellations.

Sums have broad applications in computation and analysis. They are used to obtain totals, compute expected values in probability, and appear in various areas of number theory, calculus, and mathematical analysis. Notation includes the sigma symbol, with finite or infinite bounds, and sometimes multiple indices in double sums or higher. Examples illustrate how the same operation can yield compact formulas for wide classes of sequences.