Sequence | Wikipedia audio article

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00:02:19 1 Examples and notation
00:03:54 1.1 Examples
00:06:03 1.2 Indexing
00:18:45 1.3 Defining a sequence by recursion
00:23:28 2 Formal definition and basic properties
00:23:53 2.1 Formal definition
00:26:40 2.2 Finite and infinite
00:28:28 2.3 Increasing and decreasing
00:30:08 2.4 Bounded
00:30:55 2.5 Subsequences
00:32:42 2.6 Other types of sequences
00:33:15 3 Limits and convergence
00:34:17 3.1 Formal definition of convergence
00:38:02 3.2 Applications and important results
00:40:02 3.3 Cauchy sequences
00:48:09 3.4 Infinite limits
00:49:05 4 Series
00:49:50 5 Use in other fields of mathematics
00:52:01 5.1 Topology
00:56:41 5.1.1 Product topology
00:56:53 5.2 Analysis
00:58:00 5.2.1 Sequence spaces
01:00:35 5.3 Linear algebra
01:02:36 5.4 Abstract algebra
01:04:20 5.4.1 Free monoid
01:04:50 5.4.2 Exact sequences
01:05:10 5.4.3 Spectral sequences
01:05:46 5.5 Set theory
01:07:53 5.6 Computing
01:08:31 5.7 Streams
01:09:00 6 See also
01:10:09 7 Notes
01:11:16 8 References



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SUMMARY
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order matters. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length n). The position of an element in a sequence is its rank or index; it is the natural number from which the element is the image. It depends on the context or a specific convention, if the first element has index 0 or 1. When a symbol has been chosen for denoting a sequence, the nth element of the sequence is denoted by this symbol with n as subscript; for example, the nth element of the Fibonacci sequence is generally denoted Fn.
For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...). In computing and computer science, finite sequences are sometimes called strings, words or lists, the different names commonly corresponding to different ways to represent them in computer memory; infinite sequences are called streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.