Difference between revisions of "Sequence"
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− | A '''sequence''' is an ordered list of terms. Sequences may be either [[finite]] or [[infinite]] | + | A '''sequence''' is an ordered list of terms. Sequences may be either [[finite]] or [[infinite]]. |
==Definition== | ==Definition== | ||
− | A sequence of real numbers is simply a function <math>f : \mathbb{N} \rightarrow \mathbb{R}</math>. For instance, the function <math>f(x) = x^2</math> corresponds to the sequence <math>X = (x_n) = (0, 1, 4, 9, 16, \ldots)</math>. | + | A sequence of real numbers is simply a function <math>f : \mathbb{N} \rightarrow \mathbb{R}</math>. For instance, the function <math>f(x) = x^2</math> defined on <math>\mathbb{N}</math> corresponds to the sequence <math>X = (x_n) = (0, 1, 4, 9, 16, \ldots)</math>. |
==Convergence== | ==Convergence== | ||
− | + | Intuitively, a sequence '''converges''' if its terms approach a particular number. | |
+ | |||
+ | 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>. The statement that <math>(x_n)</math> converges to <math>L</math> can be written as <math>(x_n)\rightarrow L</math>. | ||
+ | |||
+ | A classic example of convergence would be to show that <math>1/n\to 0</math> as <math>n\to \infty</math>. | ||
+ | |||
+ | '''Claim''': <math>\lim_{n\to\infty}\frac{1}{n}=0</math>. | ||
+ | |||
+ | ''Proof'': Let <math>\epsilon>0</math> be arbitrary and choose <math>N>\frac{1}{\epsilon}</math>. Then for <math>n\ge N</math> we see that | ||
+ | <center><math>n>\frac{1}{\epsilon}\implies \frac{1}{n}<\epsilon\implies \left|\frac{1}{n}-0\right|<\epsilon</math></center> | ||
+ | which proves that <math>|x_n-L|<\epsilon</math>, so <math>1/n\to 0</math> as <math>n\to \infty</math> <math>\square</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 == | ||
− | * [http://www.research.att.com/~njas/sequences/ Online Encyclopedia of Integer Sequences] | + | * [http://www.research.att.com/~njas/sequences/ Online Encyclopedia of Integer Sequences] |
== See Also == | == See Also == | ||
* [[Arithmetic sequence]] | * [[Arithmetic sequence]] | ||
* [[Geometric sequence]] | * [[Geometric sequence]] | ||
− | * [[Bolzano-Weierstrass | + | * [[Bolzano-Weierstrass Theorem]] |
{{stub}} | {{stub}} |
Latest revision as of 21:18, 13 November 2022
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
.
A classic example of convergence would be to show that as
.
Claim: .
Proof: Let be arbitrary and choose
. Then for
we see that
![$n>\frac{1}{\epsilon}\implies \frac{1}{n}<\epsilon\implies \left|\frac{1}{n}-0\right|<\epsilon$](http://latex.artofproblemsolving.com/c/b/b/cbb6bfc590019eae8754fd9dc52f97f32a42772a.png)
which proves that , so
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
This article is a stub. Help us out by expanding it.