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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, the [[Fibonacci sequence]] is perhaps the most famous example.
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A '''sequence''' is an ordered list of terms.  Sequences may be either [[finite]] or [[infinite]].
  
 
==Definition==
 
==Definition==

Revision as of 12:54, 18 May 2008

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 $f(x) = x^2$ 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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