Summation Calculator – Sigma Notation
Evaluate any sum written in sigma notation: enter an expression f(n), the lower and upper bounds, and get the total, the number of terms and a closed-form check. You can also paste a plain list of numbers to add them up.
Evaluate a Sigma Sum or Add a List of Numbers
Summation (Sigma Notation) Formula
Where:
- \(a\) — lower bound: the whole number at which the summation starts
- \(b\) — upper bound: the whole number at which the summation ends (\(b \ge a\))
- \(f(n)\) — the expression evaluated at every whole number \(n\) between \(a\) and \(b\)
- \(n\) — the index of summation, the running variable
The sigma (Σ) notation is shorthand for “evaluate \(f(n)\) at each whole number from \(a\) to \(b\), inclusive, and add all the values together.” For example, \(\sum_{n=1}^{4} n = 1 + 2 + 3 + 4 = 10\).
Summation – Calculation Example
Example: add the squares of the whole numbers from 1 to 10 — the calculator’s default expression \(f(n) = n^2\) with Lower bound 1 and Upper bound 10.
\[ \sum_{n=1}^{10} n^2 = 1^2 + 2^2 + 3^2 + \dots + 10^2 = 1 + 4 + 9 + \dots + 100 \]
Using the closed form for the sum of squares:
\[ \sum_{n=1}^{N} n^2 = \frac{N(N+1)(2N+1)}{6} = \frac{10 \times 11 \times 21}{6} = 385 \]
Result: 385, obtained by adding 10 terms. The calculator’s closed-form check confirms the same value.
How to use the calculator
- Pick the Mode: “Sigma notation Σ f(n)” to evaluate a sum defined by a formula, or “List of numbers” to add values you already have.
- Enter the Expression f(n) using the variable n — for example
n^2 + 3n,1/n,2^norn!. You can use the operators+ − * / ^ ( )and!, implicit products such as2nor3(n + 1), the functionssqrt,ln,log,sin,cos,tanandabs, and the constantspiande. - Set the Lower bound (a) and Upper bound (b). Both must be whole numbers, and the span b − a can cover at most 1,000,000 terms.
- In list mode, paste your values into Numbers instead, separated by commas, spaces, semicolons or line breaks.
- Read the results: the Sum, the Number of terms, the first terms of the series, and — in sigma mode — a Closed-form check that verifies the answer against a known formula when the expression matches one.
What is sigma notation?
Sigma notation (also called summation notation) is the standard mathematical way to write a long addition in a single line. The Greek letter Σ stands for “sum”, the value below it marks the starting index, the value above it marks the ending index, and the expression to the right describes each term. Instead of writing 1 + 4 + 9 + … + 100, you write \(\sum_{n=1}^{10} n^2\) — shorter, unambiguous, and easy to adjust by changing the bounds.
Sigma sums appear everywhere in mathematics and its applications. They define means and averages, the terms of geometric and arithmetic series, polynomial coefficients, and even the results of the factorial when combined with other operations. In statistics, finance, physics and computer science, “add up f(n) for every n in a range” is one of the most common calculations there is.
The index n is just a placeholder: \(\sum_{i=1}^{5} i\) and \(\sum_{k=1}^{5} k\) mean the same thing. What matters is the expression and the bounds. Note that the bounds do not have to start at 1 — you can sum from a negative or zero lower bound, for example \(\sum_{n=-2}^{2} n = 0\), because the terms cancel in pairs.
Many common series have closed-form formulas, which give the total without adding term by term. The calculator automatically checks for four such patterns: \(\sum n\), \(\sum n^2\), \(\sum n^3\) and the geometric series \(\sum r^n\). When your expression matches one of them, the result is verified against the closed form, which is both a sanity check and a small demonstration of how the formula works.
Common closed-form summation formulas
| Series (sum from 1 to N) | Closed form |
|---|---|
| \(\sum_{n=1}^{N} n\) | \(\dfrac{N(N+1)}{2}\) |
| \(\sum_{n=1}^{N} n^2\) | \(\dfrac{N(N+1)(2N+1)}{6}\) |
| \(\sum_{n=1}^{N} n^3\) | \(\left(\dfrac{N(N+1)}{2}\right)^2\) |
| \(\sum_{n=0}^{N} r^n\), \(r \ne 1\) | \(\dfrac{r^{N+1} - 1}{r - 1}\) |
The first formula is attributed to Gauss as a schoolboy legend: adding 1 through 100, he paired numbers from the ends (1 + 100, 2 + 99, …) to get 50 pairs of 101, giving 5050 instantly. The geometric formula is the basis of compound growth, and you can explore individual power terms with the exponent calculator.
Assumptions, limitations and tips
- Whole-number bounds only. Sigma notation with a running index makes sense for integer steps, so both the lower and upper bounds must be whole numbers. For sums over continuous ranges you need integration instead.
- Span limit. The difference between the bounds can cover at most 1,000,000 terms, so a single run cannot add more than a million values. For very long sums over a known pattern, the closed-form check tells you the total instantly anyway.
- Dividing and taking roots. The expression must be defined at every n in the range. A term like
1/nfails if the range includes 0, andsqrt(n)fails for negative n — the error message names the offending value. - Factorials. The factorial operator
!accepts whole numbers from 0 up to 170; anything larger overflows the numeric range. See the factorial calculator for standalone computations. - Trigonometry. The functions
sin,cosandtanuse radians, matching mathematical convention. - Rounding. Results are shown with up to 12 significant digits. Over very long spans of fractional terms, tiny floating-point rounding can accumulate; the closed-form check is evaluated independently and flags when the two values disagree.
- List mode. Up to 50,000 numbers can be pasted at once, separated by commas, spaces, semicolons or line breaks, with a dot as the decimal separator. List mode also reports the mean of your numbers.
Frequently Asked Questions
How do I calculate a sigma sum by hand?
List every whole number from the lower bound to the upper bound, plug each into the expression, and add the results. For \(\sum_{n=1}^{4} 2n\) the terms are 2, 4, 6, 8, so the sum is 20. For long sums, look for a pattern: pairing terms, using arithmetic or geometric series rules, or applying closed-form formulas like \(\sum_{n=1}^{N} n = \frac{N(N+1)}{2}\) avoids adding term by term.
What is the sum of the numbers from 1 to 100?
5050. Using Gauss's formula \(\sum_{n=1}^{N} n = \frac{N(N+1)}{2}\), you get \(\frac{100 \times 101}{2} = 5050\). The trick behind the formula is pairing: 1 + 100, 2 + 99, and so on give 50 pairs, each adding up to 101. For any range that does not start at 1, subtract the closed form of the part below the lower bound.
Can the sum start at 0 or a negative number?
Yes. Both bounds just have to be whole numbers, so \(\sum_{n=0}^{5} 2^n\) or \(\sum_{n=-3}^{3} n\) are valid. Watch the expression, though: a term like \(1/n\) is undefined at \(n = 0\), and \(\sqrt{n}\) fails for negative \(n\), so an error names the exact index where the problem occurs. For continuous ranges you would integrate instead.
Does it matter which letter I use for the index?
No. The index is a placeholder — \(\sum_{i=1}^{5} i^2\), \(\sum_{k=1}^{5} k^2\) and \(\sum_{n=1}^{5} n^2\) all mean the same sum of squares. What matters is the shape of the expression and the bounds. Note that the calculator expects the variable to be named n in the input field, whatever letter you might see in a textbook.
How many terms are in a sum, and why is it b − a + 1?
Both bounds are inclusive, so the count is \(b - a + 1\), not \(b - a\). From 1 to 10 there are 10 terms, not 9, because both endpoints are counted. An off-by-one error here is the most common mistake when converting a written-out sum like 3 + 4 + 5 + … + 12 back into sigma notation with bounds 3 and 12.
Can this calculator handle infinite sums?
No — an upper bound must be a finite whole number, and the span covers at most 1,000,000 terms. For an infinite series you need convergence analysis: a geometric series \(\sum r^n\) converges to \(\frac{a}{1 - r}\) only when the ratio \(|r| < 1\). As a workaround, try a very large upper bound and see whether the partial sums stabilize toward a limit.
Why is my result wrong when I add many decimal terms?
Computers store decimals in binary, so over very long sums tiny rounding errors can accumulate — the closed-form check flags when the term-by-term total drifts from the exact formula. If precision matters, scale the expression to whole numbers where possible (multiply by 100 to work in cents, for instance) or rely on a closed-form result when a standard pattern like \(\sum n\), \(\sum n^2\), \(\sum n^3\) or \(\sum r^n\) is detected.