Sigma notation packs a long sum into one line: Σi=110 i² means “add i² for every i from 1 to 10.” This calculator evaluates such sums for any expression you type, writes out the first and last terms, lists every term with a running total, and checks the answer against a closed-form formula when the sum is a classic one. Switch to Π to multiply the terms instead.
How to use the summation calculator
- Type the expression in terms of the index, for example
i^2,1/k^2,(-1)^(k+1)/korbinom(10, i). - Pick the index letter you used: i, k, n, j or x.
- Enter the lower and upper limits (whole numbers, negatives allowed).
- Choose Sum Σ or Product Π.
- Read the result on the tape. The steps show the expansion; the table lists up to the first 100 terms with running totals. Sums of up to one million terms are supported.
Sigma notation and key formulas
Sums can be split and scaled term by term: Σ(f + g) = Σf + Σg and Σ c·f = c·Σf. Classic closed forms for sums from 1 to n:
| Sum | Closed form | n = 10 |
|---|---|---|
| Σ i | n(n + 1)/2 | 55 |
| Σ i² | n(n + 1)(2n + 1)/6 | 385 |
| Σ i³ | [n(n + 1)/2]² | 3,025 |
| Σ (2i − 1) | n² | 100 |
| Σ rⁱ⁻¹ | (1 − rⁿ)/(1 − r) | depends on r |
Worked example
Evaluate Σi=110 i².
- Substitute i = 1, 2, …, 10: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
- Add them: 1 + 4 + 9 + … + 100 = 385.
- Check with the formula: 10 × 11 × 21 ÷ 6 = 2,310 ÷ 6 = 385. The average term is 38.5.
Two more results the calculator reproduces:
- Σi=1100 i = 5,050, the sum the young Carl Friedrich Gauss is said to have found by pairing 1 + 100, 2 + 99 and so on.
- Σi=010 binom(10, i) = 1,024 = 2¹⁰, because the binomial coefficients in a row of Pascal’s triangle add to a power of 2.
Working with series
Partial sums of infinite series
A partial sum shows how a series behaves as terms are added. The alternating harmonic series Σ (−1)k+1/k through k = 1,000 gives 0.6926474, approaching its limit ln 2 ≈ 0.6931472 slowly, because each term only shrinks like 1/k. By contrast, Σ 1/k² converges quickly. For a geometric series, the infinite sum has a closed form; see the geometric sequence calculator.
Products
Switching to Π multiplies the terms. Πi=1n i is n!, so Π i from 1 to 200 returns 7.88658 × 10³⁷⁴, the size of 200!. For exact factorial digits use the factorial calculator.
Accuracy
Sums are accumulated with compensated (Kahan) summation, which limits rounding error even over a million terms. Results larger than about 9 × 10¹⁵ cannot hold every integer digit in double precision and are rounded in the last places. For a list of data values rather than a formula, the sum calculator is the simpler tool, and evenly spaced terms are handled by the arithmetic sequence calculator.
Frequently asked questions
How do I read sigma notation?
In Σ from i = 1 to 10 of i², the letter under the sigma is the index, the number below is where it starts, the number on top is where it stops, and the expression to the right is evaluated at each index value. Add the results: 1 + 4 + 9 + … + 100 = 385.
What can I type in the expression?
Numbers, the index variable, + − * / ^, parentheses and factorials (!), plus functions such as sqrt, abs, ln, log, exp, sin, cos, floor and ceil, and two-argument functions binom(n, k), min, max, mod and gcd. Multiplication can be implied, as in 2i + 1 or i(i + 1).
What if the upper limit is smaller than the lower limit?
The sum has no terms, and by convention an empty sum equals 0 (an empty product equals 1). The calculator reports that rather than an error.
Can it add up an infinite series?
Not literally, but partial sums with a large upper limit show where a convergent series is heading. Σ 1/k² from 1 to 1,000,000 gives 1.6449331, within one millionth of the exact limit π²/6 ≈ 1.6449341. Divergent series, such as Σ 1/k, keep growing as the limit rises.
What are the formulas for common sums?
Σ i from 1 to n is n(n + 1)/2; Σ i² is n(n + 1)(2n + 1)/6; Σ i³ is [n(n + 1)/2]²; and the sum of the first n odd numbers, Σ (2i − 1), is n². The calculator recognizes these patterns and shows the closed form as a check.