Sigma Notation Sum Calculator

Enter an expression such as k^2 + 2*k, name the index variable, and provide inclusive lower and upper bounds. The result includes the total and the number of terms evaluated, making it useful for checking algebra homework, validating formulas, and automating repeated finite-sum calculations without expanding the entire series by hand.

Enter the term and bounds from sigma notation

Read the expression beside the sigma symbol as the term, then enter it in the expression field. Use the index printed below the sigma as the index value, the number beneath it as the lower bound, and the number above it as the upper bound. For example, a sigma with k = 1 below it, 5 above it, and k^2 + 2k to the right becomes the expression k^2 + 2*k, index k, lower 1, and upper 5. Multiplication must use an asterisk, while exponentiation uses a caret. Parentheses control grouping in the familiar way. The calculator accepts decimal and scientific-notation numbers, the constants pi and e, and common functions including abs, sqrt, sin, cos, tan, log, exp, floor, ceil, min, and max. Bounds must be integers because a standard finite sigma sum advances the index by one. Both bounds are inclusive, so a range from 1 through 5 evaluates exactly five terms. The index name is case-sensitive and must match every occurrence in the expression.

Understand how the total is evaluated

The calculator parses the expression as mathematics rather than executing it as programming code. It then substitutes each integer index value from the lower bound through the upper bound, evaluates that term, and adds it to a running total. For the term k^2 with bounds 1 and 4, the evaluated terms are 1, 4, 9, and 16, producing 30. Standard precedence applies: powers are evaluated before multiplication and division, which are evaluated before addition and subtraction. Powers associate from the right, so 2^3^2 means 2^(3^2). The total uses compensated summation to reduce avoidable floating-point loss when many decimal terms are added. Results still follow JavaScript’s finite numeric precision, so irrational constants and functions are numerical approximations rather than symbolic exact forms. Any term that becomes infinite or undefined causes a clear input error instead of returning a misleading total. This commonly catches division by zero, the square root of a negative number, or a logarithm outside its real-valued domain during the selected range.

Check, interpret, and reuse the result

The response repeats the expression, index, and inclusive bounds so the calculation can be audited without relying on surrounding context. It also reports terms_count, which should equal upper minus lower plus one, and total, the numeric value of the complete finite sum. This structure is convenient when a worksheet, application, or test harness needs to preserve both the question and its answer. To check an unfamiliar result, manually calculate the first and last terms, confirm that the index variable matches, and verify that multiplication has explicit asterisks. Parentheses are especially important when a denominator or exponent contains more than one part. The calculator intentionally limits a request to 100,000 terms and a 500-character expression, keeping browser and API execution predictable. It does not perform symbolic simplification, derive a closed-form summation formula, accept an infinite upper limit, or infer omitted multiplication signs. Use it when you need the numerical value of a concrete finite range. The browser version runs locally, while an automated API request costs $0.002.