Difference between revisions of "Sequence"

Latest revision as of 20:18, 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 $f : \mathbb{N} \rightarrow \mathbb{R}$ . For instance, the function $f(x) = x^2$ defined on $\mathbb{N}$ corresponds to the sequence $X = (x_n) = (0, 1, 4, 9, 16, \ldots)$ .

Convergence

Intuitively, a sequence converges if its terms approach a particular number.

Formally, a sequence $(x_n)$ of reals converges to $L \in \mathbb{R}$ if and only if for all positive reals $\epsilon$ , there exists a positive integer $k$ such that for all integers $n \ge k$ , we have $|x_n - L| < \epsilon$ . If $(x_n)$ converges to $L$ , $L$ is called the limit of $(x_n)$ and is written $\lim_{n \to \infty} x_n$ . The statement that $(x_n)$ converges to $L$ can be written as $(x_n)\rightarrow L$ .

A classic example of convergence would be to show that $1/n\to 0$ as $n\to \infty$ .

Claim: $\lim_{n\to\infty}\frac{1}{n}=0$ .

Proof: Let $\epsilon>0$ be arbitrary and choose $N>\frac{1}{\epsilon}$ . Then for $n\ge N$ we see that

$n>\frac{1}{\epsilon}\implies \frac{1}{n}<\epsilon\implies \left|\frac{1}{n}-0\right|<\epsilon$

which proves that $|x_n-L|<\epsilon$ , so $1/n\to 0$ as $n\to \infty$ $\square$

Monotone Sequences

Many significant sequences have their terms continually increasing, such as $(n^2)$ , or continually decreasing, such as $(1/n)$ . This motivates the following definitions:

A sequence $(p_n)$ of reals is said to be

increasing if $p_n\leq p_{n+1}$ for all $n\in\mathbb{N}$ and strictly increasing if $p_n<p_{n+1}$ for all $n\in\mathbb{N}$ ,
decreasing if $p_n\geq p_{n+1}$ for all $n\in\mathbb{N}$ and strictly decreasing if $p_n>p_{n+1}$ for all $n\in\mathbb{N}$ ,
monotone if it is either decreasing or increasing.

Resources

Online Encyclopedia of Integer Sequences

@@ Line 1: / Line 1: @@
-A '''sequence''' is an ordered list of terms.  Sequences may be either [[finite]] or [[infinite]].  In [[mathematics]] we are often interested in sequences with specific properties, the [[Fibonacci sequence]] is perhaps the most famous example.
+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
+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>.
-<math>f:\mathbb{N}\rightarrow\mathbb{R}</math>
+==Convergence==
+Intuitively, a sequence '''converges''' if its terms approach a particular number.
-The numbers <math>f(n)</math> are often denoted as <math>a_n</math> and the set <math>f(\mathbb{N})</math> is denoted as the 'sequence' <math>\left\langle a_n\right\rangle</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>. The statement that <math>(x_n)</math> converges to <math>L</math> can be written as <math>(x_n)\rightarrow L</math>.
-==Convergence==
+A classic example of convergence would be to show that <math>1/n\to 0</math> as <math>n\to \infty</math>.
-The notion of 'converging sequences' is often useful in [[Analysis|real analysis]]
-Let <math>\left\langle a_n\right\rangle</math> be a real valued sequence
+'''Claim''': <math>\lim_{n\to\infty}\frac{1}{n}=0</math>.
-Let <math>L\in\mathbb{R}</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>
-We say that '<math>\lim_{n\rightarrow\infty}a_n=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:
-or '<math>\left\langle a_n\right\rangle</math> converges to <math>L</math>' if and only if
+A sequence <math>(p_n)</math> of reals is said to be
-<math>\forall\epsilon>0</math>, <math>\exists\M\in\mathbb{N}</math> such that <math>n>M\implies |L-a_n|<\epsilon</math>
+* '''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 ==
-* [http://www.research.att.com/~njas/sequences/ Online Encyclopedia of Integer Sequences] -- A really cool math tool.
+* [http://www.research.att.com/~njas/sequences/ Online Encyclopedia of Integer Sequences]
 == See Also ==
 * [[Arithmetic sequence]]
 * [[Geometric sequence]]
-* [[Bolzano-Weierstrass theorem]]
+* [[Bolzano-Weierstrass Theorem]]
 {{stub}}

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