Difference between revisions of "Sequences"

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==Finite Sequences==
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Definition of a Sequence
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: 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.
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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.
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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.
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==Infinite sequences==
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Definition of an infinite sequence
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: An '''infinite sequence''' is a sequence with an infinite number of elements.
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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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==Discrete Functions==
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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.
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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.
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===Example 1===
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<math>q_n = n + 1</math>
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<math>q = \left \{2, 3, 4, 5, 6... \right \}</math>
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This is known as an arithmetic sequence. These will be discussed later.
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===Example 2===
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<math>c_n = \cos(n-1)</math>
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<math>c = \left \{1, 0.5403..., -0.4161..., -0.9899..., 0.2836...,...\right \}</math>
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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.
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==Recursive Functions==
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'''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.
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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.
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===Example 1===
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The sequence
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<math>p_{n+1} = p_{n} + 1, p_1 = 2</math>
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<math>p = \left \{2, 3, 4, 5... \right \} </math>
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is the same arithmetic sequence mentioned earlier. However, this time it uses a recursive definition which is essentially the same.
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===Example 2===
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This is the sequence of cosine mentioned earlier:
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<math>u_{n+1} = \cos(u_n), a_1 = 1</math>
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<math>u = \left \{0.5403..., -0.9111..., 0.6128..., 0.8180..., ... \right \}</math>
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===Example 3===
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<math>s_n = 3\times s_{n-1}</math>
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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.
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==Resources==
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===Videos===
 
[https://youtu.be/p-rc5mTDt9E Introduction to Sequences] by James Sousa
 
[https://youtu.be/p-rc5mTDt9E Introduction to Sequences] by James Sousa
  
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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
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[https://youtu.be/0aCpT3dalh8 Ex: Find the General Formula For a Sequence in Fraction Form (Arithmetic/Geometric)] by James Sousa
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[https://youtu.be/zoP3UnulRcA Geometric Sequences: A Formula for the' n - th ' Term , Part 1] by patrickJMT
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[https://youtu.be/INjRH9h3oxA Finding the nth Term of the Sequence] by Krista King
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[https://youtu.be/hc64LUtPjP0 Limits of a Sequence] by James Sousa
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[https://youtu.be/9K1xx6wfN-U Sequences - Examples showing convergence or divergence] by patrickJMT
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[https://youtu.be/0piu2uP4zr4 Finding the Limit of a Sequence, 3 more examples] by patrickJMT
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[https://youtu.be/BknXGSoCNB0 Does the sequence converge or diverge?] by Krista King
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[https://youtu.be/bvPxNhs-yrI Intro to Monotonic and Bounded Sequences, Ex 1] by patrickJMT
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[https://youtu.be/npboNfgQff8 Increasing, decreasing and not monotonic sequences] by Krista King
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[https://youtu.be/tHy3TXmZpF0 Monotonic Sequences and Bounded Sequences] by The Organic Chemistry Tutor
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[https://youtu.be/RjsyEWDEQe0 Ex: Finding Terms in a Sequence Given a Recursive Formula] by James Sousa
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[https://youtu.be/IEyojX9jzv0 Recursive Sequences] by patrickJMT
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[https://youtu.be/IFHZQ6MaG6w Recursive Formulas For Sequences] by The Organic Chemistry Tutor
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==Licensing==
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Content obtained and/or adapted from:
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* [https://en.wikibooks.org/wiki/Calculus/Definition_of_a_Sequence Definition of a sequence, Wikibooks: Calculus] under a CC BY-SA license

Latest revision as of 11:28, 29 October 2021

Finite Sequences

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 . Furthermore if we have a sequence such that then . The subscript must be a non-negative integer. Also notice that starts from one and counts up.

We can describe the terms in this sequence with a formula for all non-negative integers . So under this definition is not defined, and indeed is not in the sequence.

Infinite sequences

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 , we can say for all non-negative integers . This sequence could also be denoted as where the period of ellipses implies that this sequence is infinite.

Discrete Functions

Earlier, we defined the members of the infinite sequence as for all non-negative integers . 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

This is known as an arithmetic sequence. These will be discussed later.

Example 2

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 is defined in terms of . 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 or . 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 . 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

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:

Example 3

Notice that this time, instead of saying , we defined in terms of . This definition is still valid.


Resources

Videos

Introduction to Sequences by James Sousa

What is a Sequence? Basic Sequence Info by patrickJMT

Calculating the first terms of the sequence by Krista King


Ex: Find the General Formula For a Sequence in Fraction Form (Arithmetic/Geometric) by James Sousa

Geometric Sequences: A Formula for the' n - th ' Term , Part 1 by patrickJMT

Finding the nth Term of the Sequence by Krista King


Limits of a Sequence by James Sousa

Sequences - Examples showing convergence or divergence by patrickJMT

Finding the Limit of a Sequence, 3 more examples by patrickJMT

Does the sequence converge or diverge? by Krista King


Intro to Monotonic and Bounded Sequences, Ex 1 by patrickJMT

Increasing, decreasing and not monotonic sequences by Krista King

Monotonic Sequences and Bounded Sequences by The Organic Chemistry Tutor


Ex: Finding Terms in a Sequence Given a Recursive Formula by James Sousa

Recursive Sequences by patrickJMT

Recursive Formulas For Sequences by The Organic Chemistry Tutor

Licensing

Content obtained and/or adapted from: