Difference between revisions of "Sequence"
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==Convergence== | ==Convergence== | ||
| − | 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> | + | 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 for all positive reals %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$. |
== Resources == | == Resources == | ||
Revision as of 11:53, 18 May 2008
A sequence is an ordered list of terms. Sequences may be either finite or infinite. In mathematics we are often interested in sequences with specific properties, the Fibonacci sequence is perhaps the most famous example.
Contents
Definition
A sequence of real numbers is simply a function 
. For instance, the function 
corresponds to the sequence 
.
Convergence
Let 
be a sequence of reals. converges to 
if and only if for all positive reals %epsilon
k
n \ge k
|x_n - L| < \epsilon
(x_n)
L
L![$is called the [[limit]] of$](http://latex.artofproblemsolving.com/4/a/5/4a5d5ef67db54b5910ce49f7a8abe2e38f983e8f.png)
(x_n)
\lim_{n \to \infty} x_n$.
Resources
- Online Encyclopedia of Integer Sequences -- A really cool math tool.
See Also
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