Difference between revisions of "Sequence"
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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>. | 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>. | ||
| + | |||
| + | ==Monotone Sequences== | ||
| + | Many significant sequences are increasing or decreasing. This motivates the following definitions: | ||
| + | |||
| + | A sequence <math>(p_n)</math> of reals is said to be | ||
| + | |||
| + | * '''increasing''' if <math>p_n\leq p_{n+1}</math> for all <math>n\in\mathbb{N}</math> and '''strictly increasing''' if <math>p_n<p_{n+1}</math> for all <math>n\in\mathbb{N}</math>, | ||
| + | * '''decreasing''' if <math>p_n\geq p_{n+1}</math> for all <math>n\in\mathbb{N}</math> and '''strictly decreasing''' if <math>p_n>p_{n+1}</math> for all <math>n\in\mathbb{N}</math>, | ||
| + | * monotone if it is either decreasing or increasing. | ||
== Resources == | == Resources == | ||
Revision as of 14:06, 17 October 2012
A sequence is an ordered list of terms. Sequences may be either finite or infinite.
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 
.
Monotone Sequences
Many significant sequences are increasing or decreasing. This motivates the following definitions:
A sequence 
of reals is said to be
- increasing if
for all 
and strictly increasing if 
for all , - decreasing if
for all and strictly decreasing if 
for all , - monotone if it is either decreasing or increasing.
for all 
and strictly increasing if 
for all 
for all 
for all Resources
See Also
This article is a stub. Help us out by expanding it.
