Difference between revisions of "Sequence"
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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 14:43, 17 October 2012
A sequence is an ordered list of terms. Sequences may be either finite or infinite.
Contents
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
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