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Summation Calculator

Sigma Notation Sums With Each Term Written Out and a Running Total

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How the Summation Calculator works

This summation calculator adds up an expression for every whole value of n between two limits, which is what the Σ symbol in sigma notation asks for. Type the expression using n, such as n^2, 2n + 1 or 1/n, set where n starts and stops, and get the total with the first terms written out and a running total for each step. It handles up to 10 million terms.

Sigma notation is shorthand for a long addition. Σ n² from n = 1 to 4 means 1² + 2² + 3² + 4², which is 1 + 4 + 9 + 16 = 30. The expression after the Σ says what each term looks like, and the numbers below and above it say which values of n to use.

How the summation calculator works

The calculator substitutes each whole number from the start to the end into your expression and adds the results, using compensated addition so rounding errors do not build up over millions of terms. For common sums you can check the answer against a known formula: Σ n = n(n + 1)/2 and Σ n² = n(n + 1)(2n + 1)/6.

First example. Σ n² from 1 to 10 is 1 + 4 + 9 + … + 100 = 385, which matches 10 × 11 × 21 ÷ 6 = 385.

Second example. Σ 1/n² from 1 to 1,000,000 is about 1.644933. The infinite sum is π²/6 ≈ 1.644934, so a million terms gets within one millionth of it.

How many oranges are in a square pyramid stack 8 layers high? Each layer is a square, so the count is Σ n² from 1 to 8 = 204.

Each term of n^2 and the running total from n = 1

n Term Running total
1 1 1
2 4 5
3 9 14
4 16 30
5 25 55
6 36 91
7 49 140
8 64 204
9 81 285
10 100 385
11 121 506
12 144 650
13 169 819
14 196 1,015
15 225 1,240
16 256 1,496
17 289 1,785
18 324 2,109
19 361 2,470
20 400 2,870
21 441 3,311
22 484 3,795
23 529 4,324
24 576 4,900

Tips

FAQ

What does the Σ symbol mean?

Σ is the Greek capital sigma, standing for sum. The expression next to it is added up for each whole value of the counter, starting from the number below the Σ and ending at the number above. Σ (2n − 1) from 1 to 5 means 1 + 3 + 5 + 7 + 9 = 25.

What is the sum of the first 100 numbers?

5,050. Enter n from 1 to 100 to check. The formula n(n + 1)/2 gives 100 × 101 ÷ 2, which is the trick Gauss is said to have found as a schoolboy by pairing 1 with 100, 2 with 99 and so on.

Can it find an infinite sum?

Not directly, but you can watch a sum settle. Enter 1/2^n from 1 to 50 and the total is 1 to as many places as the screen shows. If the running total keeps growing as you raise the end value, as it does for 1/n, the infinite sum does not exist.

Why is the sum of 1/n infinite when the terms shrink to zero?

The harmonic series 1 + 1/2 + 1/3 + … grows without limit, just very slowly: the first million terms add up to only about 14.4. Group the terms as 1/3 + 1/4, then 1/5 to 1/8, and each group is more than 1/2, so the total keeps climbing forever.

See also: all Trigonometry and Calculus tools, Arithmetic, Number Theory, Algebra and Geometry, Statistics.

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