Sequences:Tails - Revision history

Lila at 16:52, 20 October 2021

2021-10-20T16:52:22Z

Lila at 16:50, 20 October 2021

2021-10-20T16:50:53Z

Lila at 16:50, 20 October 2021

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Lila at 16:48, 20 October 2021

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Lila at 16:45, 20 October 2021

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Lila at 16:43, 20 October 2021

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Lila at 16:42, 20 October 2021

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Lila at 16:41, 20 October 2021

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Lila at 16:40, 20 October 2021

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Lila at 16:36, 20 October 2021

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 </table>
 <p>Recall that for a sequence <math>(a_n)_{n=1}^{\infty}</math> that converges to the real number <math>L</math> then <math>\lim_{n \to \infty} a_n = L</math>, that is <math>\forall \varepsilon > 0</math> there exists a natural number <math>n \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid a_n - L \mid < \varepsilon</math>. For any given positive <math>\varepsilon</math> we can consider the <math>n</math>-tail of the sequence <math>(a_n)</math> to be the subsequence of <math>(a_n)</math> such that all terms in this tail are within an <math>\varepsilon</math>-distance from our limit <math>L</math>. The diagram below illustrates this concept.</p>
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png '''''The Tail of a Sequence of Real Numbers''''']</div>

← Older revision		Revision as of 16:50, 20 October 2021
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−	<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png The Tail of a Sequence of Real Numbers] ~~from mathonline.wikidot.com~~</div>	+	<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png '''''The Tail of a Sequence of Real Numbers''''']</div>
−	[http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png The Tail of a Sequence of Real Numbers] from mathonline.wikidot.com

	<p>The following theorem tells us that the m-tail of a sequence must also converge to the limit <math>L</math> provided the parent sequence <math>(a_n)</math> converges to <math>L</math>.</p>		<p>The following theorem tells us that the m-tail of a sequence must also converge to the limit <math>L</math> provided the parent sequence <math>(a_n)</math> converges to <math>L</math>.</p>

← Older revision		Revision as of 16:50, 20 October 2021
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	<p>Recall that for a sequence <math>(a_n)_{n=1}^{\infty}</math> that converges to the real number <math>L</math> then <math>\lim_{n \to \infty} a_n = L</math>, that is <math>\forall \varepsilon > 0</math> there exists a natural number <math>n \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid a_n - L \mid < \varepsilon</math>. For any given positive <math>\varepsilon</math> we can consider the <math>n</math>-tail of the sequence <math>(a_n)</math> to be the subsequence of <math>(a_n)</math> such that all terms in this tail are within an <math>\varepsilon</math>-distance from our limit <math>L</math>. The diagram below illustrates this concept.</p>		<p>Recall that for a sequence <math>(a_n)_{n=1}^{\infty}</math> that converges to the real number <math>L</math> then <math>\lim_{n \to \infty} a_n = L</math>, that is <math>\forall \varepsilon > 0</math> there exists a natural number <math>n \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid a_n - L \mid < \varepsilon</math>. For any given positive <math>\varepsilon</math> we can consider the <math>n</math>-tail of the sequence <math>(a_n)</math> to be the subsequence of <math>(a_n)</math> such that all terms in this tail are within an <math>\varepsilon</math>-distance from our limit <math>L</math>. The diagram below illustrates this concept.</p>

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		+	<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png The Tail of a Sequence of Real Numbers] from mathonline.wikidot.com</div>
	[http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png The Tail of a Sequence of Real Numbers] from mathonline.wikidot.com		[http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png The Tail of a Sequence of Real Numbers] from mathonline.wikidot.com

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 <p>Recall that for a sequence <math>(a_n)_{n=1}^{\infty}</math> that converges to the real number <math>L</math> then <math>\lim_{n \to \infty} a_n = L</math>, that is <math>\forall \varepsilon > 0</math> there exists a natural number <math>n \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid a_n - L \mid < \varepsilon</math>. For any given positive <math>\varepsilon</math> we can consider the <math>n</math>-tail of the sequence <math>(a_n)</math> to be the subsequence of <math>(a_n)</math> such that all terms in this tail are within an <math>\varepsilon</math>-distance from our limit <math>L</math>. The diagram below illustrates this concept.</p>
-{{external media
+[http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png The Tail of a Sequence of Real Numbers] from mathonline.wikidot.com
 <p>The following theorem tells us that the m-tail of a sequence must also converge to the limit <math>L</math> provided the parent sequence <math>(a_n)</math> converges to <math>L</math>.</p>

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 <p>Recall that for a sequence <math>(a_n)_{n=1}^{\infty}</math> that converges to the real number <math>L</math> then <math>\lim_{n \to \infty} a_n = L</math>, that is <math>\forall \varepsilon > 0</math> there exists a natural number <math>n \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid a_n - L \mid < \varepsilon</math>. For any given positive <math>\varepsilon</math> we can consider the <math>n</math>-tail of the sequence <math>(a_n)</math> to be the subsequence of <math>(a_n)</math> such that all terms in this tail are within an <math>\varepsilon</math>-distance from our limit <math>L</math>. The diagram below illustrates this concept.</p>
-[[File: http://mathonline.wdfiles.com/local--files/the-tail-of-a-sequence-of-real-numbers/Screen%20Shot%202014-10-13%20at%202.27.28%20AM.png|thumb|ye]]
+{{external media
 <p>The following theorem tells us that the m-tail of a sequence must also converge to the limit <math>L</math> provided the parent sequence <math>(a_n)</math> converges to <math>L</math>.</p>