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Sum
The sum of two numbers is what we get when we add the two numbers together. This operation is called summation. There are a number of ways of writing sums, with the most common being:
 Addition ([math]2+4+6 = 12[/math])
 Summation ([math]\sum_{k=1}^3 k = 1+2+3=6[/math])
 Computerization:
 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, (∑), 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) given 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
 [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]
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,
 [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. The area under a shape being defined as:
 [math]\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x[/math]
Further reading
 Nicholas J. Higham, "The accuracy of floating point summation", SIAM J. Scientific Computing 14 (4), 783–799 (1993).
Other websites
 Media related to Sum at Wikimedia Commons
 Sigma Notation on PlanetMath
 Derivation of Polynomials to Express the Sum of Natural Numbers with Exponents

