Difference between revisions of "Sequence"

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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.
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A '''sequence''' is an ordered list of terms.  Sequences may be either [[finite]] or [[infinite]].
  
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==Definition==
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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>.
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==Convergence==
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Intuitively, a sequence '''converges''' if its terms approach a particular number.
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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>.  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>.
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A classic example of convergence would be to show that <math>1/n\to 0</math> as <math>n\to \infty</math>. 
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'''Claim''': <math>\lim_{n\to\infty}\frac{1}{n}=0</math>. 
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''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
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<center><math>n>\frac{1}{\epsilon}\implies \frac{1}{n}<\epsilon\implies \left|\frac{1}{n}-0\right|<\epsilon</math></center> 
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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>
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==Monotone Sequences==
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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:
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A sequence <math>(p_n)</math> of reals is said to be
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* '''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>,
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* '''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>,
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* '''monotone''' if it is either decreasing or increasing.
  
 
== Resources ==
 
== Resources ==
* [http://www.research.att.com/~njas/sequences/ Online Encyclopedia of Integer Sequences] -- A really cool math tool.
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* [http://www.research.att.com/~njas/sequences/ Online Encyclopedia of Integer Sequences]
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== See Also ==
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* [[Arithmetic sequence]]
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* [[Geometric sequence]]
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* [[Bolzano-Weierstrass Theorem]]
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{{stub}}

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

See Also

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