About the Summation (Sigma) Calculator
This summation calculator evaluates sigma notation for you. Type the expression to be summed in terms of n — for example n^2, 2n + 1, 1/2^n or sin(n) — along with the lower and upper limits, and it adds up every term to give the total, the number of terms and the average term. The first terms are written out so you can check the pattern.
It is designed for algebra, precalculus and calculus students working with series, for checking closed-form formulas such as Σn = n(n+1)/2, and for quick numerical sums in physics, finance or programming. The running-total chart shows whether a series grows steadily or settles towards a limit, which is a useful first look at convergence.
Expressions support + − × ÷, powers (^), parentheses, factorials (n!) and functions such as sqrt, exp, log (natural log), log10, sin and cos. You may use n, i, k or j as the index variable. Sums are limited to 10,000 terms and use ordinary double-precision arithmetic.
With the default inputs, the sum σ f(n) is 385. Change any value above to recalculate instantly.
How to use the summation (sigma) calculator
- 1Type the expression to sum using n as the variable, e.g. 3n + 2.
- 2Enter the lower limit (where n starts).
- 3Enter the upper limit (where n stops).
- 4Read the sum, the number of terms and the expanded terms.
- 5Check the running-total chart to see how the sum builds up.
Formula and method
Sigma notation is shorthand for adding a sequence of terms. The index n starts at the lower limit a, increases by 1 each step, and stops at the upper limit b; each value of n is substituted into f(n) and the results are added. The calculator evaluates each term numerically and keeps a running total.
Useful closed forms for checking results: Σn from 1 to N = N(N+1)/2, Σn² = N(N+1)(2N+1)/6, Σn³ = [N(N+1)/2]², and a geometric series Σrⁿ from 0 to N = (1 − r^(N+1))/(1 − r). Infinite series can be explored by using a large upper limit and watching whether the running total levels off.
- Σ
- Summation sign (capital sigma)
- n
- Index of summation
- a, b
- Lower and upper limits of the index
- f(n)
- Expression evaluated for each n
Worked examples
Sum of squares from 1 to 10
1 + 4 + 9 + … + 100 = 385. This matches the formula N(N+1)(2N+1)/6 = 10 × 11 × 21 ÷ 6 = 385.
Sum of the first 50 odd numbers
The odd numbers 1 + 3 + 5 + … + 99 add up to 2,500, which is 50². The sum of the first N odd numbers is always N².
Geometric series Σ 1/2ⁿ, n = 1 to 20
The terms 1/2 + 1/4 + 1/8 + … halve each time, and the total after 20 terms is 1 − 1/2²⁰ ≈ 0.999999. As the upper limit grows the sum approaches 1, the value of the infinite series.
Frequently asked questions
How do you read sigma notation?+
Σ with n = 1 below and 10 above, followed by n², reads "the sum of n squared as n goes from 1 to 10". You substitute each whole number from the lower to the upper limit and add the results.
What is the formula for the sum of 1 to n?+
The sum 1 + 2 + … + n equals n(n + 1)/2. For example, 1 to 100 adds up to 100 × 101 ÷ 2 = 5,050 — enter n with limits 1 and 100 to check.
Can this calculator sum an infinite series?+
Not symbolically, but you can approximate one by using a large upper limit (up to 10,000 terms). If the running total levels off, the series converges and the total approximates its limit; if it keeps growing, it diverges.
Which functions can I use in the expression?+
You can use + − * / ^, parentheses, n! (factorial), sqrt(), exp(), log() (natural log), log10(), sin(), cos(), tan() and abs(), plus constants pi and e. Multiplication can be implicit, as in 3n.
Can the lower limit be 0 or negative?+
Yes. Any whole-number limits work as long as the upper limit is not below the lower one. Watch for terms that are undefined at n = 0, such as 1/n; the calculator will tell you where it fails.