Summation Calculator (Sigma)

Sigma (Σ) is the Greek capital letter used in mathematics to denote summation — the process of adding a sequence of numbers or expressions. The notation Σ f(n) from n=a to n=b means you evaluate the function f at every integer from a to b and add all the results together. For example, Σ n from n=1 to 4 equals 1+2+3+4 = 10.

You can enter any algebraic expression using standard operators: + (addition), - (subtraction), * (multiplication), / (division), ^ (exponentiation), and parentheses for grouping. Mathematical functions such as sqrt(), sin(), cos(), tan(), log(), exp(), and constants like pi and e are also supported. Examples include n^2, 2*n-1, sqrt(n), sin(n*pi), and 1/n.

Simple summation refers to the straightforward addition of a list of numbers (e.g. 1+2+3+4=10). Sigma notation summation is a compact mathematical way of expressing the same idea using the Σ symbol with a defined variable, lower bound, and upper bound. Sigma notation is especially powerful for expressing patterns and formulas that would be tedious to write out term by term.

To evaluate a summation, substitute each integer value of the variable (from the lower bound to the upper bound) into the expression, calculate the result for each, and add all the values together. For example, Σ n² from n=1 to 4 gives 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30. This calculator does all those steps automatically and shows each term.

Yes. Several common series have well-known closed-form results: the sum of the first n integers is n(n+1)/2, the sum of squares is n(n+1)(2n+1)/6, and the sum of cubes is [n(n+1)/2]². Geometric series with ratio r sum to a(1−rⁿ)/(1−r). This calculator evaluates finite sums numerically, which works for any expression regardless of whether a closed form exists.

The calculator supports up to 10,000 terms (i.e. the difference between upper and lower bounds can be at most 9,999). For most practical and academic purposes this is more than sufficient. If you need to evaluate infinite series or very large sums, symbolic mathematics software may be more appropriate.

Yes. You can use any single letter as your summation variable — common choices are n, i, k, and x. Just make sure the variable you declare in the Variable field matches exactly what appears in your expression. The calculator will substitute that variable for each integer value from the lower to the upper bound.

This calculator can evaluate any finite summation where the expression can be computed at each integer index. This includes arithmetic series, geometric series, polynomial series (n², n³, etc.), trigonometric series, and mixed expressions. It does not evaluate symbolic or infinite series — both bounds must be finite integers.