Summationsnotation

Summation notation, represented by the Greek letter Sigma (Σ), denotes the sum of a sequence of terms indexed by an integer variable. The general form is Σ_{i=a}^{b} f(i), where a and b are integers and f(i) is the term being summed. The sum runs over all integer values of i from a to b inclusive. If b is infinity, the notation describes an infinite series: Σ_{i=a}^{∞} f(i).

In finite sums, simple algebraic rules apply. Constants can be pulled out: Σ c·f(i) = c·Σ f(i). Sums are additive: Σ [f(i) + g(i)] = Σ f(i) + Σ g(i). The index can be shifted via a change of variables, with limits adjusted accordingly. These properties facilitate computation and manipulation of sums.

Infinite sums, or series, consider the sequence of partial sums S_n = Σ_{i=a}^{n} f(i). The series converges if S_n approaches a finite limit as n → ∞; otherwise it diverges. Convergence is analyzed with tests such as comparison, ratio, and root tests, among others, and may depend on the behavior of f(i).

Common results include geometric series: Σ_{k=0}^{∞} ar^k = a/(1−r) for |r| < 1, and the finite sum Σ_{k=0}^{n} ar^k = a(1−r^{n+1})/(1−r). The sum of the first n integers is Σ_{i=1}^{n} i = n(n+1)/2, illustrating a simple closed form. Summation notation also supports multiple indices, e.g., Σ_{i=1}^{m} Σ_{j=1}^{n} a_{ij}. Reordering or reindexing can simplify expressions, subject to convergence considerations in the infinite case.