A summation calculator evaluates a sigma expression — the compact Σ notation for "add these terms up" — without you having to write out every term. This guide explains the notation, the calculator's syntax, and the series formulas you can use to verify the result.
On this page
Reading sigma notation
The expression Σ has three parts: the summand (the term to add, written in terms of an index), the lower bound (the starting index value, written below Σ), and the upper bound (the ending index, written above Σ). For example, "the sum of x² as x runs from 1 to 10" means 1² + 2² + … + 10².
The calculator syntax
Σ(f(x), start, end)Type the summand in terms of x, then the integer start and end values. The index steps by 1 and includes both bounds.
Worked examples
Σ(x^2, 1, 10) → 385 (1² + 2² + … + 10²)
Σ(x, 1, 100) → 5050 (the famous 1 + 2 + … + 100)
Σ(2^x, 0, 10) → 2047 (2⁰ + 2¹ + … + 2¹⁰ = 2¹¹ − 1)
Σ(x^3, 1, 4) → 100 (also equals (1+2+3+4)² — no coincidence!)
Σ(1/x, 1, 5) → 2.2833… (1 + ½ + ⅓ + ¼ + ⅕)
Closed-form check formulas
| Series | Closed form | Check with |
|---|---|---|
| 1 + 2 + … + n | n(n+1)/2 | Σ(x, 1, n) |
| 1² + 2² + … + n² | n(n+1)(2n+1)/6 | Σ(x^2, 1, n) |
| 1³ + 2³ + … + n³ | [n(n+1)/2]² | Σ(x^3, 1, n) |
| 1 + r + … + rⁿ | (rⁿ⁺¹ − 1)/(r − 1) | Σ(r^x, 0, n) |
Rules & limits
- Both bounds are integers, and the start must be ≤ the end.
- The index variable in the summand is
x. - Very large ranges take longer because every term is evaluated; this is a true sum, not a formula.
Summation sits alongside integration and differentiation in COMP mode — see the pillar guide on the integration & derivative calculator, or the focused walkthrough on computing a definite integral.
Frequently asked questions
How do I evaluate a summation on the calculator?
Use Σ(f(x), start, end) — e.g. Σ(x^2, 1, 10) returns 385.
Must the bounds be integers?
Yes — the index steps through whole numbers from start to end inclusive, with start ≤ end.
What's the difference between Σ and ∫?
Σ adds discrete terms at integer indices; ∫ measures continuous area. Use Σ for series, ∫ for areas.
Add it up instantly
Open the calculator and try Σ(x^2, 1, 10) — you should get 385.