Difference between revisions of "Sequence"
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| − | A '''sequence''' is an ordered list of terms. Sequences may be either [[finite]] or [[infinite]]. In [[mathematics]] we are often interested in sequences with specific properties. | + | A '''sequence''' is an ordered list of terms. Sequences may be either [[finite]] or [[infinite]]. In [[mathematics]] we are often interested in sequences with specific properties, the [[Fibonacci sequence]] is perhaps the most famous example. |
| + | ==Definition== | ||
| + | A '''sequence of real numbers''' is simply a function | ||
| + | |||
| + | <math>f:\mathbb{N}\rightarrow\mathbb{R}</math> | ||
| + | |||
| + | The numbers <math>f(n)</math> are often denoted as <math>a_n</math> and the set <math>f(\mathbb{N})</math> is denoted as the 'sequence' <math>\left\langle a_n\right\rangle</math> | ||
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| + | ==Convergence== | ||
| + | The notion of 'converging sequences' is often useful in [[Analysis|real analysis]] | ||
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| + | Let <math>\left\langle a_n\right\rangle</math> be a real valued sequence | ||
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| + | Let <math>L\in\mathbb{R}</math> | ||
| + | |||
| + | We say that '<math>\lim_{n\rightarrow\infty}a_n=L</math>' | ||
| + | |||
| + | or '<math>\left\langle a_n\right\rangle</math> converges to <math>L</math>' if and only if | ||
| + | |||
| + | <math>\forall\epsilon>0</math>, <math>\exists\M\in\mathbb{N}</math> such that <math>n>M\implies |L-a_n|<\epsilon</math> | ||
== Resources == | == Resources == | ||
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* [[Arithmetic sequence]] | * [[Arithmetic sequence]] | ||
* [[Geometric sequence]] | * [[Geometric sequence]] | ||
| + | * [[Bolzano-Weierstrass theorem]] | ||
{{stub}} | {{stub}} | ||
Revision as of 05:39, 23 February 2008
A sequence is an ordered list of terms. Sequences may be either finite or infinite. In mathematics we are often interested in sequences with specific properties, the Fibonacci sequence is perhaps the most famous example.
Contents
Definition
A sequence of real numbers is simply a function
The numbers 
are often denoted as 
and the set 
is denoted as the 'sequence' 
Convergence
The notion of 'converging sequences' is often useful in real analysis
Let be a real valued sequence
Let 
We say that '
'
or ' converges to 
' if and only if
, $\exists\M\in\mathbb{N}$ (Error compiling LaTeX. Unknown error_msg) such that 
Resources
- Online Encyclopedia of Integer Sequences -- A really cool math tool.
See Also
This article is a stub. Help us out by expanding it.
