Difference between revisions of "Sequences:Tails"

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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''''']</div>
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| image1 = [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 Fighting style of Greek phalangites with long lances during the Roman-Spartan War] (Note the late Greek hoplite helmets with open visors made of several parts and not from one like in earlier times. The leg protection was often leather to allow for faster movement. This fighting style was not in use during the [[Battle of Marathon]]; at that time the lances were shorter and held with one hand. Longer lances, held with both hands, were adopted with the introduction of lighter hoplites and later [[phalangite]]s. As a result of their long and heavy lance which was handled with both arms they needed a lighter shield than the old hoplites.<ref>''Warfare in the Classical World'',p. 34f (Greek Hoplite (c.480BC)) p. 67 (Iphicrates reforms)</ref><ref>{{cite web |url=http://www.ancientmesopotamia.net/id27.html |title=Battle of Marathon |accessdate=2006-12-26 |work=Ancient Mesopotamia|archiveurl = https://web.archive.org/web/20060224052909/http://www.ancientmesopotamia.net/id27.html |archivedate = February 24, 2006|url-status=dead}}</ref>)
 
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<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>

Latest revision as of 10:52, 20 October 2021

The Tail of a Sequence of Real Numbers

We will now look at an important aspect of a sequence known as the tail of a sequence.

Definition: Let be a sequence of real numbers. Then for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{N}} , the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} -Tail of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} is a the subsequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_{m+1}, a_{m+2}, ... ) = (a_{m+n} : n \in \mathbb{N})} .

Recall that for a sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)_{n=1}^{\infty}} that converges to the real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} a_n = L} , that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall \varepsilon > 0} there exists a natural number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{N}} such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq N} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a_n - L \mid < \varepsilon} . For any given positive Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} we can consider the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -tail of the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} to be the subsequence of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} such that all terms in this tail are within an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} -distance from our limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} . The diagram below illustrates this concept.

The Tail of a Sequence of Real Numbers

The following theorem tells us that the m-tail of a sequence must also converge to the limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} provided the parent sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} converges to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} .

Theorem 1: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} be a sequence of real numbers. Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} converges to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} if and only if for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{N}} the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} -tail of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} , call it Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_{n_k})} converges to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} .

Resources

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