# What is the value of my summation?

Adds up a term f(n) for every whole number n from a lower to an upper limit (sigma notation), exactly as a fraction when the term allows, with the terms written out.

- Page: https://www.acalculator.org/math/summation-calculator
- JSON spec: https://www.acalculator.org/math/summation-calculator.json
- Version: 8c0e870c4f5a

## Default answer

Example with the default inputs (Term f(n) n^2, Lower limit 1, Upper limit 10): The sum of n^2 for n = 1 to 10 is 385.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Term f(n) | The term to add up, written with the index n (or i, k, j), such as n^2, 2n + 1 or 1/n. |
| a | Lower limit | The first value of the index, a whole number from -100 to 100. |
| b | Upper limit | The last value of the index, a whole number from the lower limit up to 1000. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| sum | Sum | The sum of the term over every index from the lower to the upper limit. |
| exact | Exact sum | The sum as a whole number or a fraction in lowest terms, when every term is a fraction. |
| count | Number of terms | How many values the index takes: upper limit − lower limit + 1. |
| expanded | Terms | The terms written out: all of them when there are 5 or fewer, else the first three and the last. |
| problem | Sum of | The summation in words: the term and the range of the index. |

## Method

Σ f(n) for n = a to b = f(a) + f(a + 1) + … + f(b): the term is worked out at every whole number from a to b and the values are added.

## Assumptions

- The index takes every whole number from the lower limit to the upper limit, both included; the upper limit must be at least the lower one.
- A term made of numbers, the index, + − × ÷ and whole-number powers is added up exactly in fractions (0^0 counts as 1); other terms are added in floating point with compensated summation.
- log and ln are the natural logarithm; angles are in radians.

## Worked examples

1. f = n^2, a = 1, b = 10 gives sum = 385, exact = 385, count = 10, expanded = 1 + 4 + 9 + … + 100. Source: Sigma notation as in OpenStax College Algebra 2e, section 9.4: https://openstax.org/books/college-algebra-2e/pages/9-4-series-and-their-notations.
2. f = n, a = 1, b = 100 gives sum = 5,050, count = 100.
3. f = 1/k, a = 1, b = 10 gives sum = 2.928968, exact = 7381/2520.
4. f = (1/2)^i, a = 0, b = 4 gives sum = 1.9375, exact = 31/16, expanded = 1 + 1/2 + 1/4 + 1/8 + 1/16.
5. f = 2n - 1, a = -2, b = 2 gives sum = -5, expanded = (-5) + (-3) + (-1) + 1 + 3.
6. f = sqrt(n), a = 1, b = 4 gives sum = 6.146264, expanded = 1 + 1.414213562 + 1.732050808 + 2.

## FAQ

### What does sigma notation mean?

Σ (the Greek capital sigma) means "add up". Σ n² for n = 1 to 10 means 1² + 2² + 3² + … + 10². The letter under the sigma is the index; it starts at the lower limit and goes up by 1 until it reaches the upper limit.

### How do I calculate a summation by hand?

Write out each term by putting each value of the index into the term, then add them. For Σ (2n − 1) from n = 1 to 4: 1 + 3 + 5 + 7 = 16. For long sums, use a formula, such as 1 + 2 + … + n = n(n + 1)/2.

### What are the formulas for common sums?

Σ k for k = 1 to n is n(n + 1)/2. Σ k² is n(n + 1)(2n + 1)/6. Σ k³ is (n(n + 1)/2)². A geometric sum Σ rᵏ for k = 0 to n is (1 − rⁿ⁺¹)/(1 − r) when r is not 1.

### Why does the answer show a fraction?

When every term is a fraction (the term uses only numbers, the index, + − × ÷ and whole-number powers), the calculator adds them exactly. The sum of 1/k for k = 1 to 10 is exactly 7381/2520, about 2.928968.

### Can the limits be negative?

Yes. The index can start below 0: Σ (2n − 1) for n = −2 to 2 is (−5) + (−3) + (−1) + 1 + 3 = −5. The upper limit must be at least the lower limit, and a term such as 1/n has no value at n = 0.

### Which letter should I use for the index?

Use n, i, k, or j, and use only one of them in the term. The letter does not change the sum: Σ i² and Σ n² over the same limits are equal.

### Can I sum to infinity?

No. This calculator adds a finite number of terms, with the upper limit up to 1,000. For a geometric series with |r| < 1, the infinite sum is a/(1 − r); the sequence calculator shows it.

## Sources

- OpenStax, College Algebra 2e, section 9.4 Series and Their Notations (summation notation): https://openstax.org/books/college-algebra-2e/pages/9-4-series-and-their-notations
- Wikipedia, Summation (capital-sigma notation and closed forms): https://en.wikipedia.org/wiki/Summation