en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

2014-03-28T19:02:03Z

WP:CHECKWIKI error fixes + other fixes using AWB (10065)

en>Malcolma: cat

2011-01-09T18:17:40Z

cat

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{{Orphan|date=May 2012}}

In [[mathematics]], an '''interleave sequence''' is obtained by merging two [[sequence]]s.

Let <math>S</math> be a [[Set (mathematics)|set]], and let <math>(x_i)</math> and <math>(y_i)</math>, <math>i=0,1,2,...,</math> be two [[sequence]]s in <math>S.</math> The interleave sequence is defined to be the sequence <math>x_0, y_0, x_1, y_1, \dots.</math> Formally, it is the sequence <math>(z_i), i=0,1,2,...</math> given by

:<math> z_i := \left\{\begin{matrix} x_k & \mbox{ if } i=2k \mbox{ is even,}\\ y_k & \mbox{ if } i=2k+1 \mbox{ is odd.} \end{matrix}\right.</math>

==Properties==
* The interleave sequence <math>(z_i)</math> is [[limit of a sequence|convergent]] [[if and only if]] the sequences <math>(x_i)</math> and <math>(y_i)</math> are convergent and have the same limit.

* Consider two [[real number]]s ''a'' and ''b'' greater than zero and smaller than 1. One can interleave the sequences of digits of ''a'' and ''b'', which will determine a third number ''c'', also greater than zero and smaller than 1. In this way one obtains an [[Injective function|injection]] from the square (0, 1)×(0, 1) to the [[interval (mathematics)|interval]] (0, 1).

{{PlanetMath attribution|id=449|title=Interleave sequence}}

[[Category:Real analysis]]
[[Category:Sequences and series]]

Pohlmeyer charge - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

en>Malcolma: cat