Sequence - Revision history

Vipul: /* Terminology */

2012-09-29T23:19:04Z

Terminology

← Older revision		Revision as of 23:19, 29 September 2012
Line 27:		Line 27:
	\|-		\|-
	\| index or position of a term \|\| the position in the sequence where the term occurs. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values. \|\| the index of <math>a_n</math> is <math>n</math>. \|\| the index of <math>f(n)</math> is <math>n</math>.		\| index or position of a term \|\| the position in the sequence where the term occurs. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values. \|\| the index of <math>a_n</math> is <math>n</math>. \|\| the index of <math>f(n)</math> is <math>n</math>.
			\|-
			\| successor or "next term" \|\| the successor or next term to a term is the term with index one more. In other words, the successor or "next term" to the <math>n^{th}</math> term is the <math>(n + 1)^{th}</math> term.<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition. \|\| The successor to <math>a_n</math> is <math>a_{n+1}</math> \|\| The successor to <math>f(n)</math> is <math>f(n + 1)</math>
			\|-
			\| predecessor or "previous term" \|\| the predecessor to a term is the term with index one less. In other words, the predecessor or "previous term" to the <math>n^{th}</math> term is the <math>(n - 1)^{th}</math> term.<<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.<br>The first term doesn't have a predecessor. \|\| The predecessor to <math>a_n</math> is <math>a_{n-1}</math> if <math>n > 1</math>. \|\| The predecessor to <math>f(n)</math> is <math>f(n - 1)</math>.
	\|-		\|-
	\| repetition-free sequence \|\| a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>a_m \ne a_n</math>. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>f(m) \ne f(n)</math>.		\| repetition-free sequence \|\| a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>a_m \ne a_n</math>. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>f(m) \ne f(n)</math>.
Line 39:		Line 43:
	\|-		\|-
	\| range of a sequence \|\| the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated. \|\| The set <math>\{ a_n \mid n \in \mathbb{N} \}</math> \|\| The set <math>\{ f(n) \mid n \in \mathbb{N} \}</math>		\| range of a sequence \|\| the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated. \|\| The set <math>\{ a_n \mid n \in \mathbb{N} \}</math> \|\| The set <math>\{ f(n) \mid n \in \mathbb{N} \}</math>
	\|-
	\| successor or "next term" \|\| the successor or next term to a term is the term with index one more. In other words, the successor or "next term" to the <math>n^{th}</math> term is the <math>(n + 1)^{th}</math> term.<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition. \|\| The successor to <math>a_n</math> is <math>a_{n+1}</math> \|\| The successor to <math>f(n)</math> is <math>f(n + 1)</math>
	\|-
	\| predecessor or "previous term" \|\| the predecessor to a term is the term with index one less. In other words, the predecessor or "previous term" to the <math>n^{th}</math> term is the <math>(n - 1)^{th}</math> term.<<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.<br>The first term doesn't have a predecessor. \|\| The predecessor to <math>a_n</math> is <math>a_{n-1}</math> if <math>n > 1</math>. \|\| The predecessor to <math>f(n)</math> is <math>f(n - 1)</math>.
	\|}		\|}

Vipul: /* Terminology */

2012-09-28T01:48:14Z

Terminology

← Older revision		Revision as of 01:48, 28 September 2012
Line 28:		Line 28:
	\| index or position of a term \|\| the position in the sequence where the term occurs. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values. \|\| the index of <math>a_n</math> is <math>n</math>. \|\| the index of <math>f(n)</math> is <math>n</math>.		\| index or position of a term \|\| the position in the sequence where the term occurs. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values. \|\| the index of <math>a_n</math> is <math>n</math>. \|\| the index of <math>f(n)</math> is <math>n</math>.
	\|-		\|-
	\| repetition-free sequence \|\| a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>a_m \ne a_n</math>. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>f(m) \ne f(n)</math>. \|\|		\| repetition-free sequence \|\| a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>a_m \ne a_n</math>. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>f(m) \ne f(n)</math>.
	\|-		\|-
	\| constant sequence \|\| a sequence for which the corresponding function is constant, i.e., a sequence where all terms are equal to each other. \|\| For all <math>m,n \in \mathbb{N}</math>, <math>a_m = a_n</math> \|\| For all <math>m,n \in \mathbb{N}</math>, <math>f(m) = f(n)</math>.		\| constant sequence \|\| a sequence for which the corresponding function is constant, i.e., a sequence where all terms are equal to each other. \|\| For all <math>m,n \in \mathbb{N}</math>, <math>a_m = a_n</math> \|\| For all <math>m,n \in \mathbb{N}</math>, <math>f(m) = f(n)</math>.
Line 36:		Line 36:
	\| periodic sequence \|\| a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number <math>h</math> such that for all natural numbers <math>n</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term. The smallest such <math>h</math> is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1. \|\| <math>a_n = a_{n + h} \ \forall \ n \in \mathbb{N}</math> \|\| <math>f(n) = f(n + h) \ \forall \ n \in \mathbb{N}</math>		\| periodic sequence \|\| a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number <math>h</math> such that for all natural numbers <math>n</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term. The smallest such <math>h</math> is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1. \|\| <math>a_n = a_{n + h} \ \forall \ n \in \mathbb{N}</math> \|\| <math>f(n) = f(n + h) \ \forall \ n \in \mathbb{N}</math>
	\|-		\|-
	\| eventually periodic sequence \|\| a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers <math>n_0</math> and <math>h</math> such that for all natural numbers <math>n \ge n_0</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term.		\| eventually periodic sequence \|\| a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers <math>n_0</math> and <math>h</math> such that for all natural numbers <math>n \ge n_0</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term. \|\| <math>a_n = a_{n + h}</math> for all <math>n \in \mathbb{N}</math> satisfying <math>n \ge n_0</math>\|\| <math>f(n) = f(n + h)</math> for all <math>n \in \mathbb{N}</math> satisfying <math>n \ge n_0</math>
	\|-		\|-
	\| range of a sequence \|\| the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated. \|\| The set <math>\{ a_n \mid n \in \mathbb{N} \}</math> \|\| The set <math>\{ f(n) \mid n \in \mathbb{N} \}</math>		\| range of a sequence \|\| the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated. \|\| The set <math>\{ a_n \mid n \in \mathbb{N} \}</math> \|\| The set <math>\{ f(n) \mid n \in \mathbb{N} \}</math>

Vipul: /* Terminology */

2012-09-28T01:47:24Z

Terminology

← Older revision		Revision as of 01:47, 28 September 2012
Line 34:		Line 34:
	\| eventually constant sequence \|\| a sequence for which there exists a natural number <math>n_0</math> such that the part of the sequence beyond that point is constant \|\| For <math>m,n \in \mathbb{N}</math> with <math>m,n \ge n_0</math>, <math>a_m = a_n</math> \|\| For <math>m,n \in \mathbb{N}</math> with <math>m,n \ge n_0</math>, <math>f(m) = f(n)</math>		\| eventually constant sequence \|\| a sequence for which there exists a natural number <math>n_0</math> such that the part of the sequence beyond that point is constant \|\| For <math>m,n \in \mathbb{N}</math> with <math>m,n \ge n_0</math>, <math>a_m = a_n</math> \|\| For <math>m,n \in \mathbb{N}</math> with <math>m,n \ge n_0</math>, <math>f(m) = f(n)</math>
	\|-		\|-
	\| periodic sequence \|\| a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number <math>h</math> such that for all natural numbers <math>n</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term. The smallest such <math>h</math> is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1.		\| periodic sequence \|\| a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number <math>h</math> such that for all natural numbers <math>n</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term. The smallest such <math>h</math> is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1. \|\| <math>a_n = a_{n + h} \ \forall \ n \in \mathbb{N}</math> \|\| <math>f(n) = f(n + h) \ \forall \ n \in \mathbb{N}</math>
	\|-		\|-
	\| eventually periodic sequence \|\| a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers <math>n_0</math> and <math>h</math> such that for all natural numbers <math>n \ge n_0</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term.		\| eventually periodic sequence \|\| a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers <math>n_0</math> and <math>h</math> such that for all natural numbers <math>n \ge n_0</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term.

Vipul: /* Terminology */

2012-09-28T01:46:46Z

Terminology

← Older revision		Revision as of 01:46, 28 September 2012
Line 22:		Line 22:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Term !! Meaning !! ~~In functional notation !!~~ In subscript notation for a sequence <math>(a_n)_{n \in \mathbb{N}}</math> !! In function notation for sequence given by a function <math>n</math>		! Term !! Meaning !! In subscript notation for a sequence <math>(a_n)_{n \in \mathbb{N}}</math> !! In function notation for sequence given by a function <math>n</math>
	\|-		\|-
	\| term of a sequence \|\| the appropriate function value \|\| <math>a_n</math> is the <math>n^{th}</math> term \|\| <math>f(n)</math> is the <math>n^{th}</math> term		\| term of a sequence \|\| the appropriate function value \|\| <math>a_n</math> is the <math>n^{th}</math> term \|\| <math>f(n)</math> is the <math>n^{th}</math> term

Vipul at 01:46, 28 September 2012

2012-09-28T01:46:07Z

← Older revision		Revision as of 01:46, 28 September 2012
Line 17:		Line 17:
	==Notation==		==Notation==

	Instead of using the typical function notation with the input to the function in parentheses, sequences are typically notated using a subscript notation. The sequence is named by a letter, and individual terms of the sequence are denoted by that letter with a subscript used for the position (index). For instance, for a sequence denoted with letter <math>a</math>, the first term is denoted <math>a_1</math>, the second term is denoted <math>a_2</math>, and the <math>n^{th}</math> term is denoted <math>a_n</math>.		Instead of using the typical function notation with the input to the function in parentheses, sequences are typically notated using a subscript notation. The sequence is named by a letter, and individual terms of the sequence are denoted by that letter with a subscript used for the position (index). For instance, for a sequence denoted with letter <math>a</math>, the first term is denoted <math>a_1</math>, the second term is denoted <math>a_2</math>, and the <math>n^{th}</math> term is denoted <math>a_n</math>. The sequence itself is written with the shorthand <math>(a_n)_{n \in \mathbb{N}}</math>.

	==Terminology==		==Terminology==

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Term !! Meaning		! Term !! Meaning !! In functional notation !! In subscript notation for a sequence <math>(a_n)_{n \in \mathbb{N}}</math> !! In function notation for sequence given by a function <math>n</math>
	\|-		\|-
	\| term of a sequence \|\| ~~for a sequence given by a~~ function <math>n</math>, the <math>n^{th}</math> term ~~is the value~~ <math>f(n)</math>~~. If the sequence~~ is ~~written using subscript notation, as <math>a_1,a_2,\dots</math>, then~~ the <math>n^{th}</math> term ~~is the value <math>a_n</math>.~~		\| term of a sequence \|\| the appropriate function value \|\| <math>a_n</math> is the <math>n^{th}</math> term \|\| <math>f(n)</math> is the <math>n^{th}</math> term
	\|-		\|-
	\| index or position of a term \|\| the ~~index of a~~ term ~~<math>a_n</math> is the number <math>n</math>~~. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values.		\| index or position of a term \|\| the position in the sequence where the term occurs. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values. \|\| the index of <math>a_n</math> is <math>n</math>. \|\| the index of <math>f(n)</math> is <math>n</math>.
	\|-		\|-
	\| repetition-free sequence \|\| a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct.		\| repetition-free sequence \|\| a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>a_m \ne a_n</math>. \|\| If <math>m,n \in \mathbb{N}</math> with <math>m \ne n</math>, then <math>f(m) \ne f(n)</math>. \|\|
	\|-		\|-
	\| constant sequence \|\| a sequence for which the corresponding function is constant, i.e., a sequence where all terms are equal to each other.		\| constant sequence \|\| a sequence for which the corresponding function is constant, i.e., a sequence where all terms are equal to each other. \|\| For all <math>m,n \in \mathbb{N}</math>, <math>a_m = a_n</math> \|\| For all <math>m,n \in \mathbb{N}</math>, <math>f(m) = f(n)</math>.
	\|-		\|-
	\| eventually constant sequence \|\| a sequence for which there exists a natural number <math>n_0</math> such that the part of the sequence beyond that point is constant, ~~i.e, for~~ <math>m,n \ge n_0</math>, ~~the~~ <math>m^{th}</math> ~~term and~~ <math>n~~^{th}~~</math> ~~term are equal.~~		\| eventually constant sequence \|\| a sequence for which there exists a natural number <math>n_0</math> such that the part of the sequence beyond that point is constant \|\| For <math>m,n \in \mathbb{N}</math> with <math>m,n \ge n_0</math>, <math>a_m = a_n</math> \|\| For <math>m,n \in \mathbb{N}</math> with <math>m,n \ge n_0</math>, <math>f(m) = f(n)</math>
	\|-		\|-
	\| periodic sequence \|\| a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number <math>h</math> such that for all natural numbers <math>n</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term. The smallest such <math>h</math> is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1.		\| periodic sequence \|\| a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number <math>h</math> such that for all natural numbers <math>n</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term. The smallest such <math>h</math> is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1.
Line 38:		Line 38:
	\| eventually periodic sequence \|\| a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers <math>n_0</math> and <math>h</math> such that for all natural numbers <math>n \ge n_0</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term.		\| eventually periodic sequence \|\| a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers <math>n_0</math> and <math>h</math> such that for all natural numbers <math>n \ge n_0</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term.
	\|-		\|-
	\| range of a sequence \|\| the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated.		\| range of a sequence \|\| the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated. \|\| The set <math>\{ a_n \mid n \in \mathbb{N} \}</math> \|\| The set <math>\{ f(n) \mid n \in \mathbb{N} \}</math>
	\|-		\|-
	\| successor or "next term" \|\| the successor or next term to a term is the term with index one more. In other words, the successor or "next term" to the <math>n^{th}</math> term is the <math>(n + 1)^{th}</math> term.<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.		\| successor or "next term" \|\| the successor or next term to a term is the term with index one more. In other words, the successor or "next term" to the <math>n^{th}</math> term is the <math>(n + 1)^{th}</math> term.<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition. \|\| The successor to <math>a_n</math> is <math>a_{n+1}</math> \|\| The successor to <math>f(n)</math> is <math>f(n + 1)</math>
	\|-		\|-
	\| predecessor or "previous term" \|\| the predecessor to a term is the term with index one less. In other words, the predecessor or "previous term" to the <math>n^{th}</math> term is the <math>(n - 1)^{th}</math> term.<<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.<br>The first term doesn't have a predecessor.		\| predecessor or "previous term" \|\| the predecessor to a term is the term with index one less. In other words, the predecessor or "previous term" to the <math>n^{th}</math> term is the <math>(n - 1)^{th}</math> term.<<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.<br>The first term doesn't have a predecessor. \|\| The predecessor to <math>a_n</math> is <math>a_{n-1}</math> if <math>n > 1</math>. \|\| The predecessor to <math>f(n)</math> is <math>f(n - 1)</math>.
	\|}		\|}


	There are various other notions associated with sequences specifically in the context of sequences that take values in the real numbers, i.e., sequences of real numbers.		There are various other notions associated with sequences specifically in the context of sequences that take values in the real numbers, i.e., sequences of real numbers.
Line 71:		Line 70:

	Note that the notions of bounded and convergent are special in that both notions refer to ''eventual'' behavior only. A sequence is bounded if and only if it is ''eventually'' bounded, i.e., the first few finitely many terms do not affect whether or not the sequence is bounded. Similarly, a sequence is convergent if and only if it is eventually convergent.		Note that the notions of bounded and convergent are special in that both notions refer to ''eventual'' behavior only. A sequence is bounded if and only if it is ''eventually'' bounded, i.e., the first few finitely many terms do not affect whether or not the sequence is bounded. Similarly, a sequence is convergent if and only if it is eventually convergent.

			==Sequences indexed from zero onward==

			For a number of applications, particularly [[power series]], it is useful to consider sequences that are indexed starting from zero. Such sequences can be thought of as functions from the set <math>\mathbb{N}_0 = \mathbb{N} \cup \{ 0 \} = \{ 0,1,2,\dots \}</math>. All the definitions and concepts developed for sequences can be considered for sequences indexed from zero onward.

			==Operations on sequences==

			===Pointwise operations===

			We can do various pointwise operations on sequences just as we do other types of [[pointwise combination of functions]], such as a pointwise sum, difference, product, or quotient (assuming the second sequence has no zero term). Explicitly, for sequences <math>(a_n)_{n \in \mathbb{N}}</math> and <math>(b_n)_{n \in \mathbb{N}}</math>:

			* The sum is the sequence whose <math>n^{th}</math> term is <math>a_n + b_n</math>
			* For a real number <math>\lambda</math>, <math>\lambda(a_n)</math> is the sequence whose <math>n^{th}</math> term is <math>\lambda a_n</math>.
			* The difference is the sequence whose <math>n^{th}</math> term is <math>a_n - b_n</math>
			* The product is the sequence whose <math>n^{th}</math> term is <math>a_nb_n</math>
			* The quotient is the sequence whose <math>n^{th}</math> term is <math>a_n/b_n</math>. Note that this sequence makes sense only if none of the <math>b_n</math>s are zero.

			===Shift operations===

			A shift operation on a sequence takes its terms and moves them all to the left or right.

			* A ''left shift'' operation moves all the terms a certain amount to the left. The original first few terms disappear. Explicitly, left shifting a sequence <math>(a_n)</math> by a natural number <math>h</math> means the <math>n^{th}</math> term of the new sequence is <math>a_{n + h}</math>. The original first <math>h</math> terms have disappeared in the process.
			* A ''right shift'' operation moves all the terms a certain amount to the right. This introduces the problem that the first few terms of the new sequence are undefined, and need to be specified separately. Explicitly, left shifting a sequence <math>(a_n)</math> by a natural number <math>h</math> means the <math>n^{th}</math> term of the new sequence is <math>a_{n - h}</math>. The first <math>h</math> terms of the new sequence need to be specified separately.

Vipul: /* Terminology */

2012-09-28T01:06:06Z

Terminology

@@ Line 51: / Line 51: @@
 ! Term !! Meaning
 |-
-| increasing sequence (sometimes called ''strictly'' increasing sequence) || If <math>m < n</math>, the <math>m^{th}</math> term is less than the <math>n^{th}</math> term. Note that it suffices to check that each term is smaller than the next term.
+| increasing sequence (sometimes called ''strictly'' increasing sequence to distinguish it from non-decreasing sequence) || If <math>m < n</math>, the <math>m^{th}</math> term is less than the <math>n^{th}</math> term. Note that it suffices to check that each term is less than the next term.

Vipul: Created page with "==Definition== A '''sequence''' in a set $S$ is a function from the set of natural numbers $\mathbb{N} = \{ 1,2,3,\dots \}$ to $S$ . The way ..."

2012-09-28T00:57:40Z

Created page with "==Definition== A '''sequence''' in a set <math>S</math> is a function from the set of natural numbers <math>\mathbb{N} = \{ 1,2,3,\dots \}</math> to <math>S</math>. The way ..."

New page

==Definition==

A '''sequence''' in a set <math>S</math> is a function from the set of natural numbers <math>\mathbb{N} = \{ 1,2,3,\dots \}</math> to <math>S</math>.

The way such a sequence is described is simply by listing the images of 1,2,3,... in the right order. Explicitly, for a function <math>f: \mathbb{N} \to S</math>, the sequence can be written as:

<math>f(1), f(2), f(3), f(4), f(5), \dots</math>

The values <math>f(n), n \in \mathbb{N}</math> are called the '''terms''' of the sequence. Specifically, the value <math>f(n)</math> is called the <math>n^{th}</math> term.

For instance, the sequence given by the function <math>f(n) := n^2</math> can be written as:

<math>1, 4, 9, 16, 25, 36, 49, 64, \dots</math>

Note that it is not possible to unambiguously describe a sequence (which is infinite) by just listing the first few terms (of which there are only finitely many), but the general idea behind listing the first few terms and putting the ellipses ("...") is that people are expected to figure out the most natural choice of function that fits the first few terms.

==Notation==

Instead of using the typical function notation with the input to the function in parentheses, sequences are typically notated using a subscript notation. The sequence is named by a letter, and individual terms of the sequence are denoted by that letter with a subscript used for the position (index). For instance, for a sequence denoted with letter <math>a</math>, the first term is denoted <math>a_1</math>, the second term is denoted <math>a_2</math>, and the <math>n^{th}</math> term is denoted <math>a_n</math>.

==Terminology==

{| class="sortable" border="1"
! Term !! Meaning
|-
| term of a sequence || for a sequence given by a function <math>n</math>, the <math>n^{th}</math> term is the value <math>f(n)</math>. If the sequence is written using subscript notation, as <math>a_1,a_2,\dots</math>, then the <math>n^{th}</math> term is the value <math>a_n</math>.
|-
| index or position of a term || the index of a term <math>a_n</math> is the number <math>n</math>. Note that because a sequence may have repeated terms (i.e., the function may not be one-one), a single value could occur as a term at multiple places and hence have multiple index values.
|-
| repetition-free sequence || a sequence for which the corresponding function is one-one, i.e., a sequence where all terms are distinct.
|-
| constant sequence || a sequence for which the corresponding function is constant, i.e., a sequence where all terms are equal to each other.
|-
| eventually constant sequence || a sequence for which there exists a natural number <math>n_0</math> such that the part of the sequence beyond that point is constant, i.e, for <math>m,n \ge n_0</math>, the <math>m^{th}</math> term and <math>n^{th}</math> term are equal.
|-
| periodic sequence || a sequence whose terms repeat in well defined periodic cycles, i.e., there is a natural number <math>h</math> such that for all natural numbers <math>n</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term. The smallest such <math>h</math> is termed the period of the sequence. Note that constant sequences are precisely the periodic sequences with period 1.
|-
| eventually periodic sequence || a sequence such that, ignoring the first few terms, the terms repeat in well defined periodic cycles, i.e., there are natural numbers <math>n_0</math> and <math>h</math> such that for all natural numbers <math>n \ge n_0</math>, the <math>n^{th}</math> term equals the <math>(n + h)^{th}</math> term.
|-
| range of a sequence || the range of the function defining the sequence. The range conveys information only about what values are attained. It does not store information about the ordering of the terms. It also does not store information about what terms were repeated.
|-
| successor or "next term" || the successor or next term to a term is the term with index one more. In other words, the successor or "next term" to the <math>n^{th}</math> term is the <math>(n + 1)^{th}</math> term.<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.
|-
| predecessor or "previous term" || the predecessor to a term is the term with index one less. In other words, the predecessor or "previous term" to the <math>n^{th}</math> term is the <math>(n - 1)^{th}</math> term.<<br>Note that this concept of successor depends not just on the value of the term but on its position (i.e., index). This could be a problem for sequences that have repetition.<br>The first term doesn't have a predecessor.
|}

There are various other notions associated with sequences specifically in the context of sequences that take values in the real numbers, i.e., sequences of real numbers.

{| class="sortable" border="1"
! Term !! Meaning
|-
| increasing sequence (sometimes called ''strictly'' increasing sequence) || If <math>m < n</math>, the <math>m^{th}</math> term is less than the <math>n^{th}</math> term. Note that it suffices to check that each term is smaller than the next term.

Sequence - Revision history

Vipul: /* Terminology */

Vipul: /* Terminology */

Vipul: /* Terminology */

Vipul: /* Terminology */

Vipul at 01:46, 28 September 2012

Vipul: /* Terminology */

Vipul: Created page with "==Definition== A '''sequence''' in a set S is a function from the set of natural numbers \mathbb{N} = \{ 1,2,3,\dots \} to S. The way ..."

Vipul: Created page with "==Definition== A '''sequence''' in a set $S$ is a function from the set of natural numbers $\mathbb{N} = \{ 1,2,3,\dots \}$ to $S$ . The way ..."