Summation Symbol (∑): Meaning, Uses and How to Type It

The summation symbol ∑ is a large capital sigma that means "add up." In sigma notation, ∑ᵢ₌₁ⁿ aᵢ is shorthand for a₁ + a₂ + … + aₙ: the expression after the sign is evaluated for every value of the index from the lower limit to the upper limit, and the results are summed. It is how means, series and most of statistics are written.

Below: the character to copy, the codes for every platform, how to read sigma notation, and how to write sums in LaTeX.

Summation Symbol (∑): Copy and Paste

Copy the symbol from the box below, or take the code you need from the table:

1∑
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Use ∑ for sums and Σ for the letter itself (for example a covariance matrix Σ). The letter and its other meanings are covered in the sigma symbol guide.

Summation Definition

A sum written with ∑ has four parts: the index (often i, j, k or n), the lower limit below the sign, the upper limit above it, and the summand to its right. For example:

∑ᵢ₌₁⁴ i² = 1² + 2² + 3² + 4² = 30

The upper limit can be ∞, which gives an infinite series such as ∑ₙ₌₁^∞ 1/2ⁿ = 1. Leonhard Euler introduced the capital sigma for sums in 1755, choosing S, the first letter of the Latin "summa," in its Greek form.

A few rules make sums easy to manipulate: constants factor out, ∑ c·aᵢ = c ∑ aᵢ; sums split, ∑ (aᵢ + bᵢ) = ∑ aᵢ + ∑ bᵢ; and the classic closed form ∑ᵢ₌₁ⁿ i = n(n + 1)/2.

Uses of the Summation Symbol

• Statistics: the mean is x̄ = (1/n) ∑ xᵢ, and the sample variance is s² = ∑ (xᵢ − x̄)² / (n − 1).
• Series and calculus: geometric series ∑ arⁿ = a/(1 − r) for |r| < 1, Taylor series ∑ f⁽ⁿ⁾(a)(x − a)ⁿ/n!, and Riemann sums that become integrals in the limit.
• Linear algebra and physics: matrix multiplication (AB)ᵢⱼ = ∑ₖ AᵢₖBₖⱼ, and Einstein notation, which drops the ∑ and sums over repeated indices automatically.
• Programming and spreadsheets: a ∑ is a for loop that accumulates a total; Excel's SUM function and the AutoSum button (labelled ∑) do the same.

How to Type the Summation Symbol

• Mac: press Option + W for ∑. Option + Shift + P gives the product sign ∏.
• Windows: there is no Alt code for ∑ itself. Alt + 228 types the Greek Σ, which is fine for plain text. In Word, type 2211 and press Alt + X.
• Word equations: press Alt + =, type \sum and a space, then fill in the limits; Word stacks them above and below the sign in display equations.
• Google Docs: Insert → Equation, then type \sum followed by a space.
• HTML: ∑ or ∑.
• Phones: copy ∑ from this page; the Greek keyboard gives Σ.

Summation in LaTeX

Write \sum with a subscript for the lower limit and a superscript for the upper limit. In display math the limits go above and below the sign; inline they sit to the side.

1\documentclass{article}
2\usepackage{amsmath}
3\begin{document}
4Inline: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
5
6Display:
7\[ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}, \qquad
8   \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x \]
9
10Limits forced inline: $\sum\limits_{k=1}^{n} a_k$
11
12Two conditions under the sign:
13\[ \sum_{\substack{1 \le i \le n \\ i \text{ odd}}} i = \left\lceil \frac{n}{2} \right\rceil^2 \]
14
15Double sum: $\displaystyle \sum_{i=1}^{m} \sum_{j=1}^{n} a_{ij}$
16\end{document}
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• \limits puts the limits above and below in inline math, and \nolimits does the opposite in display math. \displaystyle makes an inline sum full size.
• \substack (amsmath) stacks several conditions under the sign.
• The related big operators are \prod (∏), \coprod (∐), \bigcup and \bigcap; they take limits the same way.
• Use \sum, not \Sigma: the letter does not grow in display math or carry limits.

For sums alongside integrals and limits, see integrals, sums and limits in LaTeX. Every big operator with its command is on the summation symbol list.

Related symbols and guides

• Summation symbol page — copy the symbol and its LaTeX command
• Summation and big operators reference — every symbol in the category with its LaTeX command
• Integrals, sums and limits in LaTeX — how to typeset these symbols in a document
• Sigma symbol (Σ σ)
• Infinity symbol (∞)

Summation Symbol FAQ

What does ∑ mean in math?

∑ means "sum." ∑ᵢ₌₁ⁿ aᵢ is the total a₁ + a₂ + … + aₙ, where i runs from the lower limit 1 to the upper limit n.

Is ∑ the same as the Greek letter Σ?

They look alike but are different characters. ∑ (U+2211) is the summation operator; Σ (U+03A3) is the Greek capital sigma. Use ∑ for sums in typeset math and Σ when you mean the letter.

How do I type the summation symbol on a keyboard?

On a Mac press Option + W. On Windows use Alt + 228 for the look-alike Σ, or in Word type 2211 and press Alt + X for the true ∑.

How do I write a sum in LaTeX?

Use \sum_{i=1}^{n} a_i in math mode. Add \limits to put the limits above and below in inline text, and \substack for several conditions.

What is the sum of 1 to n?

∑ᵢ₌₁ⁿ i = n(n + 1)/2. For example, 1 + 2 + … + 100 = 100 × 101 / 2 = 5050.