Formal definition and basic properties Sequence

1 formal definition , basic properties

1.1 formal definition
1.2 finite , infinite
1.3 increasing , decreasing
1.4 bounded
1.5 subsequences
1.6 other types of sequences

formal definition , basic properties

there many different notions of sequences in mathematics, of (e.g., exact sequence) not covered definitions , notations introduced below.

formal definition

for purposes of article, define sequence function domain convex subset of set of integers. definition covers several different uses of word sequence , including one-sided infinite sequences, bi-infinite sequences, , finite sequences (see below definitions). however, many authors use narrower definition requiring domain of sequence set of natural numbers. narrower definition has disadvantage rules out finite sequences , bi-infinite sequences, both of called sequences in standard mathematical practice. many authors impose requirement on codomain of function before calling sequence, requiring set r of real numbers, set c of complex numbers, or topological space.

although sequences type of function, distinguished notationally functions in input written subscript rather in parentheses, i.e. rather f(n). there terminological differences well: value of sequence @ input 1 called first element of sequence, value @ 2 called second element , etc. also, while function abstracted input denoted single letter, e.g. f, sequence abstracted input written notation such



(

a

n



)

n
∈
a




{\displaystyle (a_{n})_{n\in a}}

, or



(

a

n


)


{\displaystyle (a_{n})}

. here domain, or index set, of sequence.

sequences , limits (see below) important concepts studying topological spaces. important generalization of sequences concept of nets. net function (possibly uncountable) directed set topological space. notational conventions sequences apply nets well.

finite , infinite

the length of sequence defined number of terms in sequence.

a sequence of finite length n called n-tuple. finite sequences include empty sequence ( ) has no elements.

normally, term infinite sequence refers sequence infinite in 1 direction, , finite in other—the sequence has first element, no final element. such sequence called singly infinite sequence or one-sided infinite sequence when disambiguation necessary. in contrast, sequence infinite in both directions—i.e. has neither first nor final element—is called bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence. function set z of integers set, such instance sequence of integers ( …, −4, −2, 0, 2, 4, 6, 8… ), bi-infinite. sequence denoted



(
2
n

)

n
=
−
∞


∞




{\displaystyle (2n)_{n=-\infty }^{\infty }}

.

increasing , decreasing

a sequence said monotonically increasing, if each term greater or equal 1 before it. example, sequence



(

a

n



)

n
=
1


∞




{\displaystyle (a_{n})_{n=1}^{\infty }}

monotonically increasing if , if an+1



≥


{\displaystyle \geq }

n ∈ n. if each consecutive term strictly greater (>) previous term sequence called strictly monotonically increasing. sequence monotonically decreasing, if each consecutive term less or equal previous one, , strictly monotonically decreasing, if each strictly less previous. if sequence either increasing or decreasing called monotone sequence. special case of more general notion of monotonic function.

the terms nondecreasing , nonincreasing used in place of increasing , decreasing in order avoid possible confusion strictly increasing , strictly decreasing, respectively.

bounded

if sequence of real numbers (an) such terms less real number m, sequence said bounded above. in less words, means there exists m such n, ≤ m. such m called upper bound. likewise, if, real m, ≥ m n greater n, sequence bounded below , such m called lower bound. if sequence both bounded above , bounded below, sequence said bounded.

subsequences

a subsequence of given sequence sequence formed given sequence deleting of elements without disturbing relative positions of remaining elements. instance, sequence of positive integers (2, 4, 6, ...) subsequence of positive integers (1, 2, 3, ...). positions of elements change when other elements deleted. however, relative positions preserved.

formally, subsequence of sequence



(

a

n



)

n
∈

n





{\displaystyle (a_{n})_{n\in \mathbb {n} }}

sequence of form



(

a


n

k





)

k
∈

n





{\displaystyle (a_{n_{k}})_{k\in \mathbb {n} }}

,



(

n

k



)

k
∈

n





{\displaystyle (n_{k})_{k\in \mathbb {n} }}

strictly increasing sequence of positive integers.

other types of sequences

some other types of sequences easy define include:

an integer sequence sequence terms integers.
a polynomial sequence sequence terms polynomials.
a positive integer sequence called multiplicative, if anm = pairs n, m such n , m coprime. in other instances, sequences called multiplicative, if = na1 n. moreover, multiplicative fibonacci sequence satisfies recursion relation = an−1 an−2.




^ cite error: named reference gaughan invoked never defined (see page).
^ edward b. saff & arthur david snider (2003). chapter 2.1 . fundamentals of complex analysis. isbn 01-390-7874-6. 
^ james r. munkres. chapters 1&2 . topology. isbn 01-318-1629-2. 
^ lando, sergei k. 7.4 multiplicative sequences . lectures on generating functions. ams. isbn 0-8218-3481-9. 
^ falcon, sergio. fibonacci s multiplicative sequence . international journal of mathematical education in science , technology. 34 (2): 310–315. doi:10.1080/0020739031000158362.