Sigma Notation

Quick Reference

Sum Formula	Closed Form	Description
$\sum _{i=1}^{n}c$	$nc$	Constant sum
$\sum _{i=1}^{n}i$	$\frac{n(n+1)}{2}$	Sum of first n positive integers
$\sum _{i=1}^n{i}^2$	$\frac{n(n+1)(2n+1)}{6}$	Sum of first n squares
$\sum _{i=1}^n{i}^3$	${\left[\frac{n(n+1)}{2}\right]}^2$	Sum of first n cubes

Definition of Sigma Notation

If $a_1,a_2,\dots ,a_n$ are real numbers, then their sum $a_1+a_2+\cdots +a_n$ can be written compactly using the capital Greek letter $\mathrm{\Sigma }$ (sigma):

$$\sum _{i=1}^n{a}_i=a_1+a_2+\cdots +a_n$$

Here i is called the index of summation, 1 is the lower limit, and n is the upper limit.

Properties of Summation

For any real numbers $a_i,b_i$ and constant c:

1. $\sum _{i=1}^{n}c=nc$
2. $\sum _{i=1}^{n}c{a}_i=c\sum _{i=1}^n{a}_i$
3. $\sum _{i=1}^n(a_i\pm b_i)=\sum _{i=1}^n{a}_i\pm \sum _{i=1}^n{b}_i$

Formulas for Special Sums