Sigma Notation

Symbol	Meaning
$\sum _{k=m}^{n}f(k)$	Sum of $f(k)$ for $k=m,m+1,\dots ,n$
$m$	Lower limit of summation
$n$	Upper limit of summation
$k$	Index of summation (dummy variable)

The Sigma Notation

A compact form of expressing a lengthy sum is the use of summation notation, also called sigma notation because it uses the Greek letter $\mathrm{\Sigma }$ (uppercase sigma, corresponding to "S" for sum).

In general, if $m$ and $n$ are integers with $m\le n$, and $f(k)$ is some formula in $k$, then

$$\sum _{k=m}^{n}f(k)=f(m)+f(m+1)+f(m+2)+\cdots +f(n-1)+f(n).$$

The numbers $m$ and $n$ are called the lower and upper limits of summation, and the letter $k$ is called the index of summation.

Example. Expand and compute ${\displaystyle \sum _{k=2}^5\frac{k}{k^2+1}}$:

\begin{aligned} \sum_{k=2}^{5}\frac{k}{k^{2}+1}&=\frac{2}{2^{2}+1}+\frac{3}{3^{2}+1}+\frac{4}{4^{2}+1}+\frac{5}{5^{2}+1}\\ &=\frac{2}{5}+\frac{3}{10}+\frac{4}{17}+\frac{5}{26}\\ &=\frac{1246}{1105}. \end{aligned}

• Any convenient letter not reserved for another purpose may be used as the index. The following all denote the same sum $1^2+2^2+\cdots +{100}^2$: $$\sum _{i=1}^{100}{i}^2,\sum _{j=1}^{100}{j}^2,\sum _{n=1}^{100}{n}^2.$$ The index of summation is a dummy variable: its name does not matter.

Frequently Asked Questions

Where does sigma notation appear in calculus?

The most important use is the **Riemann sum**: to compute the area under a curve $y=f(x)$ on $[a,b]$, divide the interval into $n$ subintervals, form the sum $\sum _{k=1}^{n}f(x_k^{\ast })\mathrm{\Delta }x$, and take the limit as $n\to \mathrm{\infty }$. This limit is the definite integral ${\int }_a^{b}f(x)dx$. You cannot read an integration chapter without understanding sigma notation.

What are some useful summation formulas?

Three formulas that appear frequently in Riemann sum calculations are: $$\sum _{k=1}^{n}k=\frac{n(n+1)}{2},\sum _{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6},\sum _{k=1}^n{k}^3={\left(\frac{n(n+1)}{2}\right)}^2.$$

Can the lower limit of summation be something other than 1?

Yes, the lower limit can be any integer, including 0 or even a negative integer. The sum simply starts at whatever value of $k$ you specify. For example, $\sum _{k=0}^n{x}^k=1+x+x^2+\cdots +x^n$ is a geometric sum starting from $k=0$.