Sigmanotation

Sigma notation, denoted by the Greek capital letter Sigma (∑), is a concise convention for expressing the sum of a sequence of terms. The general form is ∑_{i=m}^{n} f(i), which signifies the sum of f(i) as i runs from i = m to i = n. When the lower bound is 1 and the upper bound is n, it often appears as ∑_{i=1}^{n} f(i).

Examples help illustrate its use. The sum of the first n integers is written as ∑_{i=1}^{n} i = n(n+1)/2. The sum of squares is ∑_{i=1}^{n} i^2 = n(n+1)(2n+1)/6. Infinite sums, or series, are written as ∑_{k=1}^{∞} a_k and are studied for convergence, depending on the behavior of the terms a_k.

Key properties include linearity and re-indexing. Linearity states that ∑ (a_i + b_i) = ∑ a_i + ∑ b_i and ∑ c a_i = c ∑ a_i for any constant c. Re-indexing and changing the bounds can transform a sum, such as shifting the index or reversing the order with an appropriate sign change.

Sigma notation is widely used across mathematics and related fields. It underpins calculations in calculus, probability, statistics, combinatorics, and numerical methods, and it provides a compact framework for summing finite sequences as well as analyzing infinite series. The summation symbol ∑ was introduced in the 18th century by Leonhard Euler and is derived from the Greek letter sigma.