Sum Calculator
Summation (Sigma Notation) Calculator works out Result and Number of terms from Summand f(i), Index variable and Lower limit. The default figures on this page give result of 385; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
4 steps
- Σ(i = 1 to 10) i^2 has 10 terms.
- Terms: 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100
- Closed form check: sum_{i=1}^{n} i^2 = n(n+1)(2n+1)/6, evaluated as P(10) - P(0) = 385.
- Total = 385.
Termstable
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
2 sources
What the Summation (Sigma Notation) Calculator calculates
Evaluates a sigma-notation sum, a pi-notation product or the mean of f(i) as the index runs from a lower to an upper limit.
How to use this Summation (Sigma Notation) Calculator
Enter Summand f(i), Index variable, Lower limit and Upper limit. Result updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Result, Number of terms and Terms.
The method follows NIST Digital Library of Mathematical Functions, section 26.1 (sums of powers, geometric series) and Graham, Knuth & Patashnik, Concrete Mathematics, chapter 2 (Sums).
Sum of i from 1 to 10 example
With Summand f(i): i, Lower limit: 1, Upper limit: 10, the calculator returns Result: 55, Number of terms: 10. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Result, Number of terms and Terms. The inputs you can vary are Summand f(i), Index variable, Lower limit, Upper limit, Step and Operation.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
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Frequently asked questions
How to calculate estimated sum?
Evaluates a sigma-notation sum, a pi-notation product or the mean of f(i) as the index runs from a lower to an upper limit. Open “How this is worked out” beside the result to see each step for your own figures.
What is the difference between actual sum and estimated sum?
On the figures this page opens with, the result is 385. No single number fits everyone: it follows Summand f(i), Index variable and Lower limit, so enter your own above and the answer updates as you type.
What is the estimated sum of 45 and 39?
It needs Summand f(i), Index variable, Lower limit, Upper limit, Step and Operation. Nothing else is required, and the figures stay in your browser.
What information do I need for the Summation (Sigma Notation) Calculator?
You need Summand f(i), Index variable, Lower limit, Upper limit, Step, Operation and Terms to list. The page opens with default figures that give result of 385, so you can see the scale before entering your own.
What does this Summation (Sigma Notation) Calculator show?
It shows Result, Number of terms, Terms, Expanded form, Notation, Closed form used to check the answer, Smallest term and Largest term, with the working and any warnings for your entries.
Can I check how the Summation (Sigma Notation) Calculator result was calculated?
Yes. The “How this is worked out” section shows the calculation steps, and the sources section identifies the references used for the method.