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]\displaystyle{ 2+4+6 = 12 }[/math])
- Summation ([math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ i }[/math], [math]\displaystyle{ j }[/math] or [math]\displaystyle{ k }[/math][1]) along with a value, such as "[math]\displaystyle{ 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.[2]
Properties
- [math]\displaystyle{ \sum_{i=1}^n 0 = 0 }[/math]
- [math]\displaystyle{ \sum_{i=1}^n 1 = n }[/math]
- [math]\displaystyle{ \sum_{i=1}^n n = n^2 }[/math]
- [math]\displaystyle{ \sum_{i=1}^n i = \frac{n(n+1)}{2} }[/math][3]
- [math]\displaystyle{ \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} }[/math][3]
- [math]\displaystyle{ \sum_{i=1}^n i^3 = \frac{n^2 (n+1)^2}{4} }[/math][3]
- [math]\displaystyle{ \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]\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} }[/math]
The geometric series of a repeating decimal can be represented in summation. For example:
- [math]\displaystyle{ \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]\displaystyle{ \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x }[/math]
Related pages
References
- ↑ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-16.
- ↑ Weisstein, Eric W. "Sum". mathworld.wolfram.com. Retrieved 2020-08-16.
- ↑ 3.0 3.1 3.2 "Calculus I - Summation Notation". tutorial.math.lamar.edu. Retrieved 2020-08-16.
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 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