# Sum

addition of a sequence of numbers
(Redirected from Summation)

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 (${\displaystyle 2+4+6=12}$)
• Summation (${\displaystyle \sum _{k=1}^{3}k=1+2+3=6}$)
• 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 (${\displaystyle \Sigma }$ ), 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 ${\displaystyle i}$ , ${\displaystyle j}$  or ${\displaystyle k}$ [1]) along with a value, such as "${\displaystyle i=2}$ ". This tells us that the summation begins at 2, and goes up by 1 until it reaches the number on the top.[2]

## Properties

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

## Applications

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

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

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

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

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

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

## References

1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-16.
2. Weisstein, Eric W. "Sum". mathworld.wolfram.com. Retrieved 2020-08-16.
3. "Calculus I - Summation Notation". tutorial.math.lamar.edu. Retrieved 2020-08-16.

• Nicholas J. Higham, "The accuracy of floating point summation", SIAM J. Scientific Computing 14 (4), 783–799 (1993).