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 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$.
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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 <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>.
  
 
== 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.

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 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 $\lim_{n \to \infty} x_n$.

Resources

See Also

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