Sigma Notation

Summation notation allows an expression that contains a sum to be expressed in a simple, compact manner. The uppercase Greek letter sigma, Σ, is used to denote the sum of a set of numbers.

Example

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=3}^7 i^2=3^2+4^2+5^2+6^2+7^2}

Let $f$ be a function and $M,N$ are integers with $N<M$ . Then

$\sum _{i=N}^{M}f(i)=f(N)+f(N+1)+f(N+2)+\cdots +f(M)$ .

We say $N$ is the lower limit and $M$ is the upper limit of the sum.

We can replace the letter $i$ with any other variable. For this reason $i$ is referred to as a dummy variable. So

$\sum _{i=1}^{4}i=\sum _{j=1}^{4}j=\sum _{\alpha =1}^{4}\alpha =1+2+3+4$

Conventionally we use the letters $i$ , $j$ , $k$ , $m$ for dummy variables.

Example

$\sum _{i=1}^{5}i=1+2+3+4+5$

Here, the dummy variable is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} , the lower limit of summation is 1, and the upper limit is 5.

Example

Sometimes, you will see summation signs with no dummy variable specified, e.g.,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_1^4 i^3=100}

In such cases the correct dummy variable should be clear from the context.

You may also see cases where the limits are unspecified. Here too, they must be deduced from the context.

Common summations

$\sum _{i=1}^{n}c=c+c+\cdots +c=nc\ ,\ c\in \mathbb {R}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^n i=1+2+3+\cdots+n=\frac{n(n+1)}{2}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^n i^2=1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}}

$\sum _{i=1}^{n}i^{3}=1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(\sum _{i=1}^{n}i\right)^{2}=\left({\frac {n(n+1)}{2}}\right)^{2}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i=1}^\infty a_i = \lim_{t \to \infty} \left[\sum_{i=1}^t a_i\right]}

Resources

• Summation notation, Wikibooks: Calculus