# Sum

addition of a sequence of numbers

The sum of two numbers is their value added together. This operation is called additive summation or addition. There are many ways of writing sums, including:

• Addition ($2+4+6=12$ )
• Summation ($\sum _{k=1}^{3}k=1+2+3=6$ )
• Code:
Sum = 0
For I = M to N
Sum = Sum + X(I)
Next I (in Visual BASIC)

## Sigma notation

Sigma notation is a mathematical notation to write long sums in a short way. Sigma notation uses the Greek letter Sigma ($\Sigma$ ), and takes upper and lower bounds which tell us where the sum begins and where it ends. The lower bound usually has a variable (called the index, often denoted by $i$ , $j$  or $k$ ) along with a value, such as "$i=2$ ". This tells us that the summation begins at 2, and goes up by 1 until it reaches the number on the top.

## Properties

$\sum _{i=1}^{n}0=0$
$\sum _{i=1}^{n}1=n$
$\sum _{i=1}^{n}n=n^{2}$
$\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}$ 
$\sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}$ 
$\sum _{i=1}^{n}i^{3}={\frac {n^{2}(n+1)^{2}}{4}}$ 
$\sum _{i=1}^{\infty }a_{i}=\lim _{t\to \infty }\sum _{i=1}^{t}a_{i}$

## Applications

Sums are used to represent series and sequences. For example:

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

The geometric series of a repeating decimal can be represented in summation. For example:

$\sum _{i=1}^{\infty }{\frac {3}{10^{i}}}=0.333333...={\frac {1}{3}}$

The concept of an integral is a limit of sums, with the area under a curve being defined as:

$\lim _{n\to \infty }\sum _{i=1}^{n}f(x_{i}^{*})\Delta x$