Difference between revisions of "Sequence"
m (→See Also) |
m (added some more terminology) |
||
Line 9: | Line 9: | ||
Formally, a sequence <math>(x_n)</math> of reals converges to <math>L \in \mathbb{R}</math> if and only if for all positive reals <math>\epsilon</math>, there exists a positive integer <math>k</math> such that for all integers <math>n \ge k</math>, we have <math>|x_n - L| < \epsilon</math>. | Formally, a sequence <math>(x_n)</math> of reals converges to <math>L \in \mathbb{R}</math> if and only if for all positive reals <math>\epsilon</math>, there exists a positive integer <math>k</math> such that for all integers <math>n \ge k</math>, we have <math>|x_n - L| < \epsilon</math>. | ||
− | If <math>(x_n)</math> converges to <math>L</math>, <math>L</math> is called the [[limit]] of <math>(x_n)</math> and is written <math>\lim_{n \to \infty} x_n</math>. | + | If <math>(x_n)</math> converges to <math>L</math>, <math>L</math> is called the [[limit]] of <math>(x_n)</math> and is written <math>\lim_{n \to \infty} x_n</math>. The statement that <math>(x_n)</math> converges to <math>L</math> can be written as <math>(x_n)\rightarrow L</math>. |
== Resources == | == Resources == |
Revision as of 13:43, 17 October 2012
A sequence is an ordered list of terms. Sequences may be either finite or infinite.
Contents
[hide]Definition
A sequence of real numbers is simply a function . For instance, the function defined on corresponds to the sequence .
Convergence
Intuitively, a sequence converges if its terms approach a particular number.
Formally, a sequence of reals converges to if and only if for all positive reals , there exists a positive integer such that for all integers , we have .
If converges to , is called the limit of and is written . The statement that converges to can be written as .
Resources
See Also
This article is a stub. Help us out by expanding it.