## Sequence of real numbers

A sequence of real numbers (or a real sequence) is defined as a function $ f: \mathbb{N} \to \mathbb{R}$ , where $ \mathbb{N}$ is the set of natural numbers and $ \mathbb{R}$ is the set of real numbers. Thus, $ f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a function which produces a sequence of real numbers $ r_n$ . It’s customary to write a sequence as form of functions in brackets, e.g.; $ \langle f(n) \rangle$ , $ \{ f(n) \}$ . We can alternatively represent a sequence as the function with natural numbers as subscripts, e.g., $ \langle f_n \rangle$ , $ \{ f_n \}$ . This alternate method is a better representation of a sequence as it distinguishes ‘a sequence’ from ‘a function’. We shall use $ \langle f_n \rangle$ notation and when writen $ \langle f_n \rangle$ , we mean $ \langle f_1, f_2, f_3, \ldots, f_n, \ldots \rangle$ a sequence with infinitely many terms. Since all of $ \{ f_1, f_2, f_3, \ldots, f_n, \ldots \}$ are real numbers, this kind of sequence is called a sequence of real numbers.

## Examples of Sequences

- Like $ f(x)=\dfrac{1}{x} \forall x \in \mathbb{R}$ is a real-valued-function, $ f(n)=\dfrac{1}{n} \forall n \in \mathbb{N}$ is a real sequence.

Putting consecutive values of $ n \in \mathbb{N}$ in $ f(n)=\dfrac{1}{n}$ we obtain a real-sequence

*n=1 f(1)=1*

*n=2 f(2)=1/2*

*n=3 f(3)=1/3*

… * …*

*n=n f(n)=1/n*

…* …*

This real-sequence can be represented by

$ \langle \dfrac{1}{n} \rangle := \langle 1, \dfrac{1}{2}, \dfrac{1}{3}, \ldots, \dfrac{1}{n}, \ldots$ .

- $ \langle {(-1)}^n \rangle$ is the sequence $ \langle -1, 1, -1, 1, \ldots, {(-1)}^n, \ldots \rangle$ .
- $ \langle -3n \rangle$ is the sequence $ \langle -3, -6, -9, \ldots, -3n, \ldots \rangle$
- A sequence can also be formed by a recurrence relation with boundary values. If $ f_n= f_{n-1}+f_{n-2} \ \text{for} n \ge 2$ and $ f_0=f_1=1$ , then we obtain the sequence $ \langle f_n \rangle$ as

*n=1 * $ f_1=1$ (given)

*n=2 * $ f_2=f_1 +f_0=1+1=2$ (given $ f_0=1=f_1$ )

*n=3 * $ f_3=f_2+f_1=2+1=3$

*n=4 * $ f_4=f_3+f_2=3+2=5$

and so on…

This sequence, $ \langle 1, 1,2, 3, 5, 8, 13, 21, \ldots \rangle$ is a real-sequence known as *Fibonacci Sequence.*

*Range Set of a Sequence*

The set of all *‘distinct’ elements *of a sequence is called the range set of the given sequence.

For example:

- The range set of $ \langle \dfrac{1}{n}\rangle:= \{ \dfrac{1}{n} : n \in \mathbb{N} \}$ , which is an infinite set.
- The range set of $ \langle {(-1)}^n \rangle := \{ -1, 1 \}$ , a finite set.

*Remark:* The range set of a sequence may be either infinite or finite, but a sequence has always an infinite number of elements.

## Sub-sequence of the Sequence

A sub-sequence of the sequence is *another sequence *containing *some *of the values of the sequence in the same order as in the original sequence. Alternatively, a sub-sequence of the sequence is another sequence which range set is a subset of the range set of the sequence.

For example:

*<1, 3, 5, 7, …>* is a sub-sequence of the sequence *<1, 2, 3, 4, …>.*
*<1, 5, 13, 21, …> *is a sub-sequence of the sequence *<1,1,2,3,5,8,13,21, 34, …>.*
*<1,1,1,1,1,…> *is a sub-sequence of the sequence *<-1, 1, -1, 1, …>.* Since, the sequence *<1,1,1,1,…> * has only one value for each term, it’s called a* ***constant sequence****.**

*Remark:* A sub-sequence is also a sequence hence it satisfy and follow all the properties of a sequence.

## Equality of two sequences

Two sequences $ \langle S_n \rangle$ and $ \langle T_n \rangle$ are said to be equal, if and only if $ S_n=T_n, \forall n \in \mathbb{N}$ .

For example: The sequences $ \langle \dfrac{n+1}{n} \rangle$ and $ \langle 1+\dfrac{1}{n} \rangle$ are equal to each other.

**Remark:**** **From the *definition* the sequences *<-1,1,-1,1, …> *and *<1,-1,1,-1,…> *are not equal to each other, though they look alike and has same range set.

## Algebra of Sequences

Let $ \langle S_n \rangle$ and $ \langle T_n \rangle$ be two sequence, then the sequences having *n*-th terms $ S_n+T_n, \ S_n-T_n, \ S_n \cdot T_n, \ \dfrac{S_n}{T_n}$ (respectively) are called the SUM, DIFFERENCE, PRODUCT, QUOTIENT of $ \langle S_n \rangle$ and $ \langle T_n \rangle$ .

*For example: *The sequence <1, 8, 19,30, …> is the sum of sequences <0, 1, 2, 3, …> and <1, 7, 17, 27, …> obtained after adding n-th term of one sequence to corresponding n-th term of other sequence. Similarly, other operations can be carried.

If $ S_n \ne 0 \forall n$ , then the sequence $ \langle \dfrac{1}{S_n} \rangle$ is known as the *reciprocal *of the sequence $ \langle S_n \rangle$ .

*For example: *$ \langle \dfrac{1}{1}, \dfrac{-1}{2}, \dfrac{1}{3}, \ldots \rangle$ is the reciprocal of the sequence $ \langle 1, -2, 3, \ldots \rangle$ .

**Remark:** The sequences *<-1,1,-1,1, …> *and *<1,-1,1,-1,…> *have their reciprocals equal to the original sequence, hence these are called* identity-sequences.*

If $ c \in \mathbb{R}$ then the sequence with *n*-th term $ cS_n$ is called the scalar multiple of sequence $ \langle S_n \rangle$ . This sequence is denoted by $ \langle cS_n \rangle$ .

## Bounds of a Sequence

- A sequence $ \langle S_n \rangle$ is said to be
**bounded above**, if there exists a real number *M *such that $ S_n \le M, \forall n \in \mathbb{N}$ . *M *is called an upper bound of the sequence $ \langle S_n \rangle$ .
- A sequence $ \langle S_n \rangle$ is said to be
**bounded below**, if there exists a real number *m *such that $ S_n \ge m, \forall n \in \mathbb{N}$ . *m *is called a lower bound of the sequence $ \langle S_n \rangle$ .
- A sequence $ \langle S_n \rangle$ is said to be
**bounded***,* if it is both bounded above and bounded below. Thus, if $ \langle S_n \rangle$ is a bounded sequence, there exist two real numbers *m & M* such that $ m \le S_n \le M \forall n \in \mathbb{N}$ .
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