Sigma Notation Calculator

Sigma notation is shorthand for 'add up these numbers.' The calculator takes an expression, an index variable, and a range of whole numbers, then evaluates the expression at every integer in that range and adds the results.

The common mistake is treating the index variable as if it matters outside the sum. It does not. Rename n to k, shift the bounds, the sum stays the same. The calculator enforces this by letting you pick any index letter (n, i, j, k) and treating it only as the stepping value inside the expression.

To use it, enter an expression using your chosen index, set a lower limit and an upper limit, and press Calculate Sum. The calculator returns the total sum, a breakdown of individual terms, a running total, and a closed form if one exists.

• Expression (Summand): The formula using the index variable, e.g., n, n^2, 2^n. Supports +, -, *, /, ^, parentheses, factorial (n!), sqrt, sin, cos, tan, ln, log, exp, abs, pi, e.
• Index Variable: n, i, j, or k. A dummy variable that does not appear in the final sum.
• Lower Limit: Starting integer value for the index. May be negative.
• Upper Limit: Ending integer value for the index. Must be a whole number. Limits over 10,000 terms are rejected for performance.

Type your expression exactly as you would in math class. Use n for the index (or change it to i, j, or k in the Index Variable box). The calculator reads n^2, 2^n, n(n+1), and n! correctly. Implicit multiplication works: 2n means 2 × n, and n(n+1) means n × (n+1).

Negative exponents and unary minus: The expression -n^2 means -(n^2), not (-n)^2. The calculator follows standard precedence: exponent before minus. If you want the square of a negative number, use parentheses: (-n)^2.

Lower and upper limits are whole numbers. They can be negative.This is not an error; it is the convention used in mathematics.

Optional series helpers let you pick a type (Arithmetic, Geometric, Power, Factorial) and fill in parameters, then press 'Apply to Expression' to write the expression for you. The calculator always evaluates whatever is in the expression box, regardless of the series type selector.

After calculation, the calculator shows several numbers. The Sum (Σ) is the total. The Number of Terms is upper bound minus lower bound plus one. The Average Value is sum divided by number of terms. The First Term and Last Term are f(lower) and f(upper).

Individual Terms table lists each index value, the expression's value at that index, and the running sum so far. This is the raw data behind the total. If you enable 'Show Terms Table', the calculator displays every row up to the limit you set (10, 20, 50, 100, or All).

Running Total (cumulative sum) shows how the sum builds term by term. A graph visualises either individual term values or the cumulative sum. Use this to see whether later terms dominate the total or whether positive and negative terms cancel.

Closed-Form Formula appears when the calculator detects a standard series: sum of integers, squares, cubes, arithmetic series, or geometric series. The closed form gives the same total as term by term addition, but in one step. For example, the sum from 1 to 10 of n² is 385. The closed form is n(n+1)(2n+1)/6 with n=10, which equals 10×11×21/6 = 385. If the expression has no standard closed form, the calculator shows 'Custom Expression' and the sum is computed term by term.

Series Type Detection

The calculator identifies the series type (Arithmetic, Geometric, Power, Constant, Custom) from the terms themselves. Arithmetic series have a constant difference between consecutive terms. Geometric series have a constant ratio. Power series have terms like n^k. Custom means no standard pattern was detected.

Three examples show how the calculator handles different expression types.

Example 1: Σ from n=1 to 5 of n

Expression: n. Lower: 1, Upper: 5. Terms: 1, 2, 3, 4, 5. Sum: 15. The calculator detects an arithmetic series (common difference 1) and shows the closed form n(n+1)/2 with n=5: 5×6/2 = 15.

Example 2: Σ from n=1 to 4 of n²

Expression: n^2. Lower: 1, Upper: 4. Terms: 1, 4, 9, 16. Sum: 30. The calculator detects a power series (n²) and shows the closed form n(n+1)(2n+1)/6 with n=4: 4×5×9/6 = 30.

Example 3: Σ from n=0 to 3 of 2ⁿ

Expression: 2^n. Lower: 0, Upper: 3. Terms: 1, 2, 4, 8. Sum: 15. The calculator detects a geometric series (ratio 2) and shows the closed form a(1‑rⁿ)/(1‑r) with a=1, r=2, N=4: 1×(1‑16)/(1‑2) = 15.

These examples come from the same formulas documented in OpenStax Precalculus 2e, section 11.4 Series and Their Notations.

Expression	Lower Limit	Upper Limit	Terms (first 3)	Sum	Closed Form (if any)
n	1	5	1, 2, 3	15	n(n+1)/2 = 15
n²	1	4	1, 4, 9	30	n(n+1)(2n+1)/6 = 30
2ⁿ	0	3	1, 2, 4	15	a(1‑rⁿ)/(1‑r) = 15
(-1)ⁿ	1	4	-1, 1, -1	0	None (alternating series)
n - 5	0	4	-5, -4, -3	-5	Arithmetic: N/2×(first+last) = -5

A zero sum does not mean the calculator failed. It means positive and negative terms cancelled exactly. Example: Σ from n=1 to 4 of (-1)ⁿ produces terms -1, 1, -1, 1, sum = 0. This is common for alternating series like (-1)ⁿ or sin(n).

A negative total means the negative terms outweighed the positive ones. Example: Σ from n=0 to 4 of (n - 5) gives terms -5, -4, -3, -2, -1, sum = -15. The expression is negative across the whole range. This is not an error; the calculator respects the sign of each term.

Empty sum: If the lower limit is greater than the upper limit (e.g., n=5 to n=2), the sum has no terms and equals 0. This is the mathematical convention, not a bug. The calculator states 'empty sum' in its output.

Very large sums: When the total exceeds 2⁵³ (about 9×10¹⁵), the calculator shows it in scientific notation. The value is approximate due to floating point limits. This affects factorial series and high powers like n¹⁰.

The calculator also refuses to evaluate more than 10,000 terms. If your range is larger, use a closed form if available, or split the range into smaller pieces.