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 have their terms continually increasing, such as <math>(n^2)</math>, or continually decreasing, such as <math>(1/n)</math>. 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 15:07, 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 have their terms continually increasing, such as , or continually decreasing, such as . 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.
Resources
See Also
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