Enter an expression in the formula field (e.g. n^2), set your lower limit and upper limit, choose your index variable, and the Sigma Notation Calculator computes the exact sum (Σ). Supports operators like +, -, *, /, ^ and functions like sqrt, sin, cos, tan, exp and constants pi and e.
Σ Sum
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Sigma notation (Σ) is a mathematical shorthand for expressing the sum of a sequence of terms. The Greek letter sigma (Σ) indicates summation, with a lower limit below it and an upper limit above it defining the range of the index variable. For example, Σ(n=1 to 5) n² means 1² + 2² + 3² + 4² + 5² = 55.
You can use standard arithmetic operators (+, -, *, /, ^) and functions like sqrt, sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, and exp. Constants pi and e are also supported. Always use the index variable you selected (n, i, k, j, or m) as the variable in your expression.
By convention, if the lower limit exceeds the upper limit, the sum is considered empty and equals 0. Most calculators, including this one, will return 0 in that case. Adjust your limits so the lower value is less than or equal to the upper value to get a meaningful result.
A finite sum has a specific upper limit (e.g., n = 1 to 100), meaning there are a countable number of terms to add. An infinite series has no upper bound and may or may not converge to a finite value. This calculator handles finite sums only — set a numeric upper limit to evaluate the summation.
Set the expression to 'i', choose 'i' as the index variable, enter 1 as the lower limit and 10 as the upper limit, then click calculate. The result is 1+2+3+…+10 = 55, which matches the closed-form formula n(n+1)/2 = 10×11/2 = 55.
Sigma notation appears throughout statistics (calculating means, variances, standard deviations), physics (summing forces or energies), finance (computing compound interest schedules), and computer science (algorithm complexity analysis). It provides a compact way to express repetitive addition patterns.
NaN (Not a Number) usually means the expression contains an invalid operation for some index value, such as dividing by zero (1/n when n=0) or taking the square root of a negative number. Check your expression and adjust your lower limit to avoid problematic values.
Start with simple expressions like n, n^2, or 2^n and compare the calculator's output to known formulas. For example, Σ n from 1 to n equals n(n+1)/2, and Σ n² from 1 to n equals n(n+1)(2n+1)/6. Verifying these identities is an excellent way to build intuition.