Difference between revisions of "Sequence"

Revision as of 11:48, 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

Let $(x_n)$ be a sequence of reals. $(x_n)$ converges to $L \in \mathbb{R}$ if and only if $\forall \epsilon > 0, \exists k \in \mathbb{N} : \forall n \in \mathbb{N}, n \ge k, |x_n - L| < \epsilon$ . If $(x_n)$ converges to $L$ , $L$ is called the limit of $(x_n)$ and is written $\lim_{n \to \infty} x_n$ .

Resources

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

@@ Line 5: / Line 5: @@
 ==Convergence==
-The notion of 'converging sequences' is often useful in [[Analysis|real analysis]]
+Let <math>(x_n)</math> be a sequence of reals. <math>(x_n)</math> '''converges''' to <math>L \in \mathbb{R}</math> if and only if <math>\forall \epsilon > 0, \exists k \in \mathbb{N} : \forall n \in \mathbb{N}, n \ge k, |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>.
-Let <math>\left\langle a_n\right\rangle</math> be a real valued sequence
-Let <math>L\in\mathbb{R}</math>
-We say that '<math>\lim_{n\rightarrow\infty}a_n=L</math>'
-or '<math>\left\langle a_n\right\rangle</math> converges to <math>L</math>' if and only if
-<math>\forall\epsilon>0</math>, <math>\exists\M\in\mathbb{N}</math> such that <math>n>M\implies |L-a_n|<\epsilon</math>
 == Resources ==

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Difference between revisions of "Sequence"

Revision as of 11:48, 18 May 2008

Contents

Definition

Convergence

Resources

See Also