# Science:Infinite Series Module/Units/Unit 1/1.2 Sigma Notation/1.2.02 Sigma Notation Terminology

## Terminology

The infinite sum

${\displaystyle a_{1}+a_{2}+a_{3}+\ldots }$

can be written in a more compact way. Using the capital greek letter sigma, ${\displaystyle \Sigma }$, we may write this sum as

${\displaystyle a_{1}+a_{2}+a_{3}+\ldots =\sum _{k=1}^{\infty }a_{k}}$

We often use the following terminology when using this notation

• the general term of the sum is ${\displaystyle a_{k}}$
• the index of summation is ${\displaystyle k}$
• the limits of summation are 1 and ${\displaystyle \infty }$, although in general the limits can be any integer, ${\displaystyle -\infty }$, or ${\displaystyle +\infty }$.

## Simple Example

Consider the infinite series

${\displaystyle {\frac {1}{2+3^{2}}}+{\frac {2}{2+4^{2}}}+{\frac {3}{2+5^{2}}}+\ldots =\sum _{k=3}^{\infty }{\frac {k-2}{2+k^{2}}}}$

In this example, the general term is

${\displaystyle {\frac {k-2}{2+k^{2}}},}$

the index of summation is ${\displaystyle k}$, and the limits of summation are 3 and infinity.