Difference between revisions of "Sequence"

Revision as of 11:41, 18 May 2008

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.

Definition

A sequence of real numbers is simply a function $f : \mathbb{N} \rightarrow \mathbb{R}$ . For instance, the function $f(x) = x^2$ corresponds to the sequence $X = (x_n) = (0, 1, 4, 9, 16, \ldots)$ .

Convergence

The notion of 'converging sequences' is often useful in real analysis

Let $\left\langle a_n\right\rangle$ be a real valued sequence

Let $L\in\mathbb{R}$

We say that ' $\lim_{n\rightarrow\infty}a_n=L$ '

or ' $\left\langle a_n\right\rangle$ converges to $L$ ' if and only if

$\forall\epsilon>0$ , $\exists\M\in\mathbb{N}$ (Error compiling LaTeX. Unknown error_msg) such that $n>M\implies |L-a_n|<\epsilon$

Resources

Online Encyclopedia of Integer Sequences -- A really cool math tool.

@@ Line 2: / Line 2: @@
 ==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> corresponds to the sequence <math>X = (x_n) = (0, 1, 4, 9, 16, \ldots)</math>.
-<math>f:\mathbb{N}\rightarrow\mathbb{R}</math>
-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>
 ==Convergence==

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Difference between revisions of "Sequence"

Revision as of 11:41, 18 May 2008

Contents

Definition

Convergence

Resources

See Also