Sequences - Revision history

Lila: /* Resources */

2021-10-29T17:28:48Z

Resources

Lila: /* Infinite sequences */

2021-10-07T22:44:19Z

Infinite sequences

Lila: /* Finite Sequences */

2021-10-07T22:40:37Z

Finite Sequences

Lila at 22:40, 7 October 2021

2021-10-07T22:40:01Z

James.kercheville: Added additional video links

2020-09-11T15:43:32Z

Added additional video links

James.kercheville: Added additional video links

2020-09-09T20:47:43Z

Added additional video links

James.kercheville: Added video links

2020-08-28T18:02:33Z

Added video links

New page

[https://youtu.be/p-rc5mTDt9E Introduction to Sequences] by James Sousa

[https://youtu.be/Kxh7yJC9Jr0 What is a Sequence? Basic Sequence Info] by patrickJMT

[https://youtu.be/XdoqvWxTVEw Calculating the first terms of the sequence ] by Krista King

← Older revision		Revision as of 17:28, 29 October 2021
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	[https://youtu.be/IFHZQ6MaG6w Recursive Formulas For Sequences] by The Organic Chemistry Tutor		[https://youtu.be/IFHZQ6MaG6w Recursive Formulas For Sequences] by The Organic Chemistry Tutor
		+
		+	==Licensing==
		+	Content obtained and/or adapted from:
		+	* [https://en.wikibooks.org/wiki/Calculus/Definition_of_a_Sequence Definition of a sequence, Wikibooks: Calculus] under a CC BY-SA license

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 ==Infinite sequences==
-{{Calculus/Def
+Definition of an infinite sequence
-|title=Definition: Definition of an infinite sequence
+: An '''infinite sequence''' is a sequence with an infinite number of elements.
 Infinite sequences have infinite terms. For such a sequence, we can again give a formula for any term in the sequence. For our previous sequence <math>a</math>, we can say <math>a_n = n</math> for all non-negative integers <math>n</math>. This sequence could also be denoted as <math>\left \{0,1,2,3,4,5,... \right \}</math> where the period of ellipses implies that this sequence is infinite.

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 ==Finite Sequences==
-{{Calculus/Def
+Definition of a Sequence
-|title=Definition: Definition of a Sequence
+: A '''sequence''' is an ordered collection of terms in which repetition is allowed. The number of terms in a sequence is called the '''length''' of the sequence.
 Sequences are often denoted by brackets like <math>\left \{2,3,3,4,4\right \}</math>. Furthermore if we have a sequence <math>a</math> such that <math>a = \left \{1,2,3,4,5\right \}</math> then <math>a_1 = 1, a_2=2, a_3=3...</math>. The subscript must be a non-negative integer. Also notice that <math>n</math> starts from one and counts up.
 We can describe the terms in this sequence with a formula <math>a_n=n</math> for all non-negative integers <math>n < 6</math>. So under this definition <math>a_6</math> is not defined, and indeed <math>a_6</math> is not in the sequence.
 ==Infinite sequences==

← Older revision		Revision as of 22:40, 7 October 2021
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		+	==Finite Sequences==
		+
		+	{{Calculus/Def
		+	\|title=Definition: Definition of a Sequence
		+	\|text=
		+	A '''sequence''' is an ordered collection of terms in which repetition is allowed. The number of terms in a sequence is called the '''length''' of the sequence.
		+	}}
		+
		+	Sequences are often denoted by brackets like <math>\left \{2,3,3,4,4\right \}</math>. Furthermore if we have a sequence <math>a</math> such that <math>a = \left \{1,2,3,4,5\right \}</math> then <math>a_1 = 1, a_2=2, a_3=3...</math>. The subscript must be a non-negative integer. Also notice that <math>n</math> starts from one and counts up.
		+
		+	We can describe the terms in this sequence with a formula <math>a_n=n</math> for all non-negative integers <math>n < 6</math>. So under this definition <math>a_6</math> is not defined, and indeed <math>a_6</math> is not in the sequence.
		+
		+	==Infinite sequences==
		+
		+	{{Calculus/Def
		+	\|title=Definition: Definition of an infinite sequence
		+	\|text=
		+	An '''infinite sequence''' is a sequence with an infinite number of elements.
		+	}}
		+
		+	Infinite sequences have infinite terms. For such a sequence, we can again give a formula for any term in the sequence. For our previous sequence <math>a</math>, we can say <math>a_n = n</math> for all non-negative integers <math>n</math>. This sequence could also be denoted as <math>\left \{0,1,2,3,4,5,... \right \}</math> where the period of ellipses implies that this sequence is infinite.
		+
		+	==Discrete Functions==
		+
		+	Earlier, we defined the members of the infinite sequence <math>a</math> as <math>a_n = n</math> for all non-negative integers <math>n</math>. This is known as a '''discrete function''', '''discrete definition''', or '''explicit definition'''. A discrete function is any function whose domain is not the set of all real or imaginary numbers, but is instead a smaller, countable set like the set of all integers or the set of all rational numbers. Note that a set differs from a sequence, but that is beyond the scope of this discussion.
		+
		+	Discrete functions only take “countable”, discrete domains. The set of all integers is countable, because there are not infinitely many values between two values in the set; there is no extra value between 2 and 1, as 1.5 is not an integer and is not contained in the set. Also note that given a discrete function or explicit definition, as long as the domain is discrete, the range must also be discrete. This means that if the input of a discrete function is countable, the output must also be countable.
		+
		+	===Example 1===
		+
		+	<math>q_n = n + 1</math>
		+
		+	<math>q = \left \{2, 3, 4, 5, 6... \right \}</math>
		+
		+	This is known as an arithmetic sequence. These will be discussed later.
		+
		+	===Example 2===
		+	<math>c_n = \cos(n-1)</math>
		+
		+	<math>c = \left \{1, 0.5403..., -0.4161..., -0.9899..., 0.2836...,...\right \}</math>
		+
		+	This result may be interesting: a sequence does not need to be a collection of integers, indeed it can be any collection, as long as it is countable. Here, we are simply taking the cosine of all integers, and any discrete function must have both a discrete domain ''and'' range.
		+
		+	==Recursive Functions==
		+
		+	'''Recursive functions''', '''recursive formulas''', or '''recursive definitions''' are formulas in which <math>a_{n}</math> is defined in terms of <math>a_{n-1}</math>. Knowing any term in a recursively defined sequence requires you to know all the terms before it, which means you must know the first term, sometimes denoted <math>a_0</math> or <math>a_1</math>. The first term must be defined in order to have a proper recursive sequence; it cannot be assumed that the first term is 1.
		+
		+	Sometimes, one can have a sequence that is ''necessarily'' defined by a recursive function. For instance, the recursively defined sequence <math>u_{n+1} = \cos(u_n), a_1 = 1</math>. This sequence cannot be expressed any other "easy" way and in this kind of situation it is best to use the recursive definition.
		+
		+	===Example 1===
		+
		+	The sequence
		+
		+	<math>p_{n+1} = p_{n} + 1, p_1 = 2</math>
		+
		+	<math>p = \left \{2, 3, 4, 5... \right \} </math>
		+
		+	is the same arithmetic sequence mentioned earlier. However, this time it uses a recursive definition which is essentially the same.
		+
		+	===Example 2===
		+
		+	This is the sequence of cosine mentioned earlier:
		+
		+	<math>u_{n+1} = \cos(u_n), a_1 = 1</math>
		+
		+	<math>u = \left \{0.5403..., -0.9111..., 0.6128..., 0.8180..., ... \right \}</math>
		+
		+	===Example 3===
		+
		+	<math>s_n = 3\times s_{n-1}</math>
		+
		+	Notice that this time, instead of saying <math>s_{n+1} = ...</math>, we defined <math>s_n</math> in terms of <math>s_{n-1}</math>. This definition is still valid.
		+
		+
		+
		+	==Resources==
		+	===Videos===
	[https://youtu.be/p-rc5mTDt9E Introduction to Sequences] by James Sousa		[https://youtu.be/p-rc5mTDt9E Introduction to Sequences] by James Sousa

← Older revision		Revision as of 15:43, 11 September 2020
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	[https://youtu.be/BknXGSoCNB0 Does the sequence converge or diverge?] by Krista King		[https://youtu.be/BknXGSoCNB0 Does the sequence converge or diverge?] by Krista King
		+
		+
		+	[https://youtu.be/bvPxNhs-yrI Intro to Monotonic and Bounded Sequences, Ex 1] by patrickJMT
		+
		+	[https://youtu.be/npboNfgQff8 Increasing, decreasing and not monotonic sequences] by Krista King
		+
		+	[https://youtu.be/tHy3TXmZpF0 Monotonic Sequences and Bounded Sequences] by The Organic Chemistry Tutor
		+
		+
		+	[https://youtu.be/RjsyEWDEQe0 Ex: Finding Terms in a Sequence Given a Recursive Formula] by James Sousa
		+
		+	[https://youtu.be/IEyojX9jzv0 Recursive Sequences] by patrickJMT
		+
		+	[https://youtu.be/IFHZQ6MaG6w Recursive Formulas For Sequences] by The Organic Chemistry Tutor

← Older revision		Revision as of 20:47, 9 September 2020
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	[https://youtu.be/XdoqvWxTVEw Calculating the first terms of the sequence ] by Krista King		[https://youtu.be/XdoqvWxTVEw Calculating the first terms of the sequence ] by Krista King
		+
		+
		+	[https://youtu.be/0aCpT3dalh8 Ex: Find the General Formula For a Sequence in Fraction Form (Arithmetic/Geometric)] by James Sousa
		+
		+	[https://youtu.be/zoP3UnulRcA Geometric Sequences: A Formula for the' n - th ' Term , Part 1] by patrickJMT
		+
		+	[https://youtu.be/INjRH9h3oxA Finding the nth Term of the Sequence] by Krista King
		+
		+
		+	[https://youtu.be/hc64LUtPjP0 Limits of a Sequence] by James Sousa
		+
		+	[https://youtu.be/9K1xx6wfN-U Sequences - Examples showing convergence or divergence] by patrickJMT
		+
		+	[https://youtu.be/0piu2uP4zr4 Finding the Limit of a Sequence, 3 more examples] by patrickJMT
		+
		+	[https://youtu.be/BknXGSoCNB0 Does the sequence converge or diverge?] by Krista King