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Sequence

A sequence is a special case of a function whose domain is the set of all natural numbers

The range of the sequence is the set of the values

Since the domain of all the sequences is the some set

We shall for the sake of simplicity and clarity, denote

where

For example, consider the sequence

then we find that even terms are of equal values, viz.

and

**N**and the range is any set, i.e. a function**ƒ : N****S**is called a sequence. If the range of sequence is any subset of real numbers, then the sequence is called a real sequence.**∴**A real sequence is a function from the set N of natural numbers into the set R of real numbers. In this aspect we shall be dealing with real numbers only, therefore, we shall use the term sequence to denote a real sequence.**∴**A sequence is a rule which associates to each natural number a unique real number.**∀ n ϵ N**there exist**ƒ(n) ϵ R**, where**ƒ(n)**denotes the real number associated by**ƒ**to n**ϵ N**.The range of the sequence is the set of the values

**{ƒ(1), ƒ(2), ….. , ƒ(n), … , } ⊂ R.**Since the domain of all the sequences is the some set

**N**of natural numbers, therefore, a sequence is determined fully if we know**ƒ(n), ∀ n ϵ N.**We shall for the sake of simplicity and clarity, denote

**ƒ(n)**by**ƒ**_{n}and the sequence itself as**< ƒ**_{n}>

∴ < ƒ∴ < ƒ

_{n}> = <ƒ_{1}, ƒ_{2}, …. , ƒ_{k}, …. , ƒ_{n}, …. >where

**ƒ**are real numbers and are called first term, second term_{1}, ƒ_{2}, …., ƒ_{n}, …**, …. N**^{th}term of the sequence. The terms**ƒ**_{1}and**ƒ**_{2}are called distinct terms of the sequence**i ≠ j**, though**ƒ**and_{i}**ƒ**are of equal value._{j}For example, consider the sequence

**< ƒ**_{n}>, where ƒ_{n}= (-1)^{n-1}, ∀ n ϵ Nthen we find that even terms are of equal values, viz.

**ƒ(1) = ƒ(3) = ƒ(5) = …. = 1, 1 ≠ 3 ≠ 5 ≠ ….**and

**ƒ(2) = ƒ(4) = ƒ(6) = …. = -1, 2 ≠ 4 ≠ 6 ≠ ….****Services: -**Sequence Homework | Sequence Homework Help | Sequence Homework Help Services | Live Sequence Homework Help | Sequence Homework Tutors | Online Sequence Homework Help | Sequence Tutors | Online Sequence Tutors | Sequence Homework Services | SequenceSubmit Your Query ???

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Topics

Real Number Absolute Value
Addition Of Matrices
Square Matrix Adjoint
Algebraic Structures
Alternating Series
Linear Equations Determinants
Archimedean Real Numbers
Binary Operation
Binary Relation In A Set
Bounded, Unbounded Sets
Cauchy Root Test
Caylay Hamiltion Theorem
Circular Permutation
Common Roots
Complex Numbers
Complex Number Conjugate
Conjugate Of A Matrix
Constant Sequences
Convergence Of A Sequence
Cosets
Cubic, Biquadratic Equations
De Moivre Theorem
Real Number Denseness
Order 3 Determinants
Differences Of Matrices
Direct Sum Of Vector Subspaces
Eigen Vector
Elementary Matrices
Matrix Elem. Transformations
Equal Matrices
Equal Roots
Two Permutations Equity
Equivalent Matrices
Trigonometry Function Expansion
Field
Function
Algebra Fundamental Theorem
Gaussian Integer
Geometric Series
Group
Ideals
Quantity Increasing Roots
Infinite Series Convergent
Integers
Inverse Of Square Matrix
Inverses Of Elementary Matrices
Iota, Imaginary Numbers
Left-Right Identity
Sequence Limit Points
Linear Combination Vectors Span
Linear Dependence, Independence
Linear Homogeneous Equations
Two Subspaces Linear Sum
Matric Polynomial
Matrix
Linear Equation Matrix Inverse
Matrix Multiplication
Matrix Scalar Multiplication
Method Of Difference
Minors And Co-factors
Multiplication Modulo P
Normal Sub-Group
Normalizer Or Centalizer
Orbit Of Permutation
Peano Axioms
Permutation Function
Pigeon Hole Principle
Matrices Integral Powers
Mathematical Induction Principal
Two Determinants Product
Two Permutations Product
Properties Of Modulus
Rank Of A Matrix
Rational Numbers
Rational, Integral Polynomial
Reciprocal Roots
Relation Of Sets
Rings Of A Set
Row By Column Matrix
Sequence
Series
Series Of Positive Terms
Series Partial Sum Sequence
Subrings
Sum Of A Series
Cosine Series Sum
Sum Of Sine Series
Symmetric/Skew Symm. Matrices
Roots Symmetric Functions
Symmetric Set Degree N
Synthetic Division
Transformation In General
Transformations Of Equations
Transpose Of A Matrix
Matrix Transposed Conjugate
Transposition
Complex Numbers Representation
Vector Space
Vector Sub-spaces
Whole Numbers

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