Sigma Notation

A sum written in one line

Long sums take a long time to write out, so mathematics has a short way to write them. The symbol $\sum$Σ is the Greek capital letter sigma, which is their S, and S stands for sum. $\sum$Σ is an instruction: add up the terms that follow.

The sum $\sum _{n=1}^{4}n$Σ n from n = 1 to 4 means: let n be 1, then 2, then 3, then 4, and add up the values. That is 1 + 2 + 3 + 4 = 10. The letter n is the counter. It is called n here, but any letter can be used; r and k are common.

Each value of n adds a block of n dots to the total: 1, then 1 + 2 = 3, then 3 + 3 = 6, then 6 + 4 = 10.

The expression names each term

The expression written after the $\sum$Σ says what each term is. In $\sum _{n=1}^{4}2n$Σ 2n from n = 1 to 4, put each value of n into 2n: 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6 and 2 × 4 = 8. Then add them: 2 + 4 + 6 + 8 = 20.

Change the expression and the terms change, even though n still runs from 1 to 4. For example, $\sum _{n=1}^4{n}^2$Σ n² from n = 1 to 4 is 1 + 4 + 9 + 16 = 30.

Each value of n from 1 to 4 goes into 2n, which makes the four terms 2, 4, 6 and 8. Their sum is 20.

The limits

In print, the start is written below the $\sum$Σ and the stop is written above it. These two numbers are the limits. In $\sum _{n=1}^{4}2n$Σ 2n from n = 1 to 4, the lower limit n = 1 says where n starts and the upper limit 4 says where it stops. Both limits are included, so n takes the values 1, 2, 3 and 4, and the sum has four terms.

The number of terms is the upper limit minus the lower limit, plus 1. For example, $\sum _{n=3}^{6}2n$Σ 2n from n = 3 to 6 has 6 − 3 + 1 = 4 terms: 6 + 8 + 10 + 12 = 36. The usual mistake is to leave out the plus 1 and count only 6 − 3 = 3 terms.

Changing the limits

Raise the upper limit by one and the sum gains exactly one term. The sum $\sum _{n=1}^{4}n$Σ n from n = 1 to 4 is 10. The sum $\sum _{n=1}^{5}n$Σ n from n = 1 to 5 adds the term for n = 5, so it is 10 + 5 = 15.

A sum that starts partway along is the difference of two sums that both start at 1. For example, $\sum _{n=3}^{5}n$Σ n from n = 3 to 5 is $\sum _{n=1}^{5}n$Σ n from n = 1 to 5 take away $\sum _{n=1}^{2}n$Σ n from n = 1 to 2, which is 15 − 3 = 12. Check by adding: 3 + 4 + 5 = 12. Notice that the sum taken away stops at 2, one below the new lower limit, because the term for n = 3 belongs to the answer.

Raising the upper limit from 4 to 5 adds one more block, the 5 dots for n = 5: the sum goes from 10 to 15.