Sum

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

Properties

<math>\sum_{i=1}^n 0 = 0</math>

<math>\sum_{i=1}^n 1 = n</math>

<math>\sum_{i=1}^n n = n^2</math>

<math>\sum_{i=1}^n i = \frac{n(n+1)}{2}</math>

<math>\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}</math>

<math>\sum_{i=1}^n i^3 = \frac{n^2 (n+1)^2}{4}</math>

<math>\sum_{i=1}^\infty a_i = \lim_{t \to \infty} \sum_{i=1}^{t} a_i</math>

Applications

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

<math>\sum_{i=1}^4 \frac{1}{2^i} = \frac{1}{2^1} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4}</math>

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

<math>\sum_{i=1}^\infty \frac{3}{10^i} = 0.333333... = \frac{1}{3}</math>

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

<math>\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x</math>

Calculating the mean of a set of numbers can be expressed like this, where each <math>x</math> can be any number.

<math>\frac{1}{|m-n|}\sum_{i=m}^n x_i</math>

So, because <math>\sum_{i=m}^n \frac{i}{2}=\frac{1}2\sum_{i=m}^n i</math>, you could also calculate the mean like <math>\sum_{i=m}^n \frac{x_i}{|m-n|}</math> as well.

Related pages

• Product (mathematics)

Further reading

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

Other websites

• File:Commons-logo.svg Media related to at Wikimedia Commons
• Sigma Notation Archived 2015-09-21 at the Wayback Machine on PlanetMath
• Derivation of Polynomials to Express the Sum of Natural Numbers with Exponents Archived 2013-02-18 at the Wayback Machine