Sigma Notation Calculator
This sigma notation calculator evaluates any summation exactly: write the expression in k, give the two limits, and every term is worked out as a fraction and added up — then added again from the far end as a check.
Start with the one everybody meets first: Σk from 1 to 100. Ten terms or a hundred, the instruction is the same — put 1 in for k, then 2, then 3, and add. The answer is 5050, and there is a shortcut for it, but the shortcut is not what the notation means.
What Sigma Notation Actually Says
The symbol looks forbidding and means something very plain. Σ is an instruction to add, and it comes with three pieces of information:
The letter underneath — usually k, sometimes i or n — is the index, and the number beside it is where the index starts. The number on top is where it stops. The expression to the right is what you write the index into.
So Σk² from k = 1 to 5 says: work out 1², then 2², then 3², 4², 5², and add them. That is 1 + 4 + 9 + 16 + 25 = 55. Nothing more mysterious than that.
This matters because the notation is often taught alongside the shortcut formulas, and the two get muddled. How to calculate sigma notation is always the same thing — substitute and add. The formulas are conveniences for particular expressions, not the meaning.
The Standard Formulas, and When They Apply
For a few common expressions there is a closed form, and they are worth knowing because they turn a hundred additions into one line:
| Σ1 | n |
| Σk | n(n + 1) / 2 |
| Σk² | n(n + 1)(2n + 1) / 6 |
| Σk³ | [n(n + 1) / 2]² |
Two conditions are easy to forget. First, they assume the index starts at 1 — a sum from k = 3 needs the first two terms taking off, or simply adding out directly. Second, they only cover polynomials in k. There is no such formula for Σ1/k, and none for Σ2k of this kind either.
This calculator applies the formula where one exists and prints it beside the term-by-term total, so you can see the two agree. Where none exists it says so plainly rather than inventing one. And because a polynomial can be written many ways, it works out the coefficients from the expression itself — k(k+1) and k²+k get the same treatment.
When a Sigma Sum Is Really a Series You Know
Here is the thing that trips people up, and it is worth saying directly: sigma notation is a way of writing a sum, not a kind of sum.
Write Σ(2k−1) from 1 to 10 and you get 1 + 3 + 5 + … + 19 — which is an arithmetic series. Write Σ2k and you get a geometric one. The notation has not changed what the sum is; it has only changed how it is written down.
So when the terms turn out to rise by a fixed step, this page says so and points you at the arithmetic series calculator, which pairs the terms up and shows where its formula comes from. When they multiply by a fixed ratio, it points at the geometric series calculator instead. Those pages do that one job properly; this one covers everything else.
The check is made over every term, not the first few — a run that looks arithmetic at the start and then turns round is not an arithmetic series, and saying it was would send you to a formula that gives a different answer.
How to Use This Sigma Notation Calculator
Three boxes: the expression, and the two limits.
The expression is written in k. 3k and 3(k+1) mean multiplication; k^2 is a power; / divides. The √ key writes sqrt(, and it only works where the root comes out exactly — a rounded root would not be an exact answer, so it is refused instead. abs() works too, typed with your own keyboard.
The limits are whole numbers, and the lower one may be 0 or negative. That is worth saying because the standard formulas assume a start of 1 and many calculators assume it too; here the terms are simply added out.
Press = and you get the total, the terms with a running total beside each, the formula where one applies, and the name of the series when there is one.
One thing this page does not do: it will not convert a written-out sum into sigma notation. It evaluates the notation, it does not compose it.
What It Refuses, and Why That Is the Point
A calculator that misreads an expression does not fail loudly — it hands you a confident number that happens to be wrong. That is the worst thing a tool like this can do, so this one refuses anything it cannot read for certain, and says what it could not read and where.
k2 is refused rather than guessed at: it might mean 2k or k², and picking one would be inventing your intention. 1/(k−3) summed across k = 3 is refused with the index named, because that term divides by zero. sqrt(k) from 2 is refused because √2 is not exact and a decimal standing in for it would be a lie about precision. An upside-down range is caught, and it suggests the likely cause — two numbers typed the wrong way round.
Everything is exact throughout. Σ1/k from 1 to 5 comes out as 137/60, not 2.2833, and a sum of fractions never drifts as the list grows.
Sigma Notation FAQ
What does sigma notation mean?
How do you calculate sigma notation by hand?
What is the formula for the sum of k from 1 to n?
Can the index start at 0 or at a negative number?
Is a sigma sum the same as a series?
Why does this calculator refuse some expressions?
Substitute, add, check. The sigma is only shorthand for the first two, and this page does the third as well — every total is added twice, and where a formula exists it has to agree before you see anything.