Real Sequences
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}$ .
- The least real number M, if exists, of the set of all upper bounds of $ \langle S_n \rangle$ is called the least upper bound (supremum) of the sequence $ \langle S_n \rangle$ .
- The greatest real number m , if exits, of the set of all lower bounds of $ \langle S_n \rangle$ is called the greatest lower bound (infimum) of the sequence $ \langle S_n \rangle$ .
Remark: If the range set of a sequence is finite, then the sequence is always bounded.
Examples:
- The sequence $ \langle n^3 \rangle := \langle 1, 8, 27, \ldots \rangle$ is bounded below by 1, but is not bounded above.
- The sequence $ \langle \dfrac{1}{n} \rangle := \langle 1, \dfrac{1}{2}, \dfrac{1}{3}, \ldots$ is bounded as it has the range set (0, 1], which is finite.
- The sequence $ \langle {(-1)}^n \rangle := \langle -1, 1, -1, \ldots$ is also bounded.
Convergent Sequence
A sequence $ \langle S_n \rangle$ is said to converge to a real number l if for each $ \epsilon$ >0, there exists a positive integer m depending on $ \epsilon$ , such that $ |S_n-l|$ < $ \epsilon \ \forall n \ge m$ .
This number l is called the limit of the sequence $ \langle S_n \rangle$ and we write this fact as $ \lim_{n \to \infty} S_n=l$ and the sequence itself is called a convergent sequence. From now on, we’ll use $ \lim S_n=l$ to represent $ \lim_{n \to \infty} S_n=l$ , unless stated.
Important Theorems on Convergent Sequences and Limit
- ( Uniqueness Theorem ) Every convergent sequence has a unique limit.
- For a sequence $ \langle S_n \rangle$ of non-negative numbers, $ \lim S_n \ge 0$ .
- Every convergent sequence is bounded, but the converse is not necessarily true.
- Let $ \lim S_n= l$ and $ T_n=l’$ , then $ \lim (S_n +T_n) = l+l’$ , $ \lim (S_n -T_n) = l-l’$ and $ \lim S_n \cdot T_n = l \cdot l’$ .
- Let $ \langle S_n \rangle$ and $ \langle T_n \rangle$ be two sequences such that $ S_n \le T_n$ , then $ \lim S_n \le \lim T_n$ .
- If $ \langle S_n \rangle$ converges to l , then $ \langle |S_n| \rangle$ converges to | l |. In other words, if $ \lim S_n = l$ then $ \lim |S_n| =|l|$ .
- ( Sandwitch Theorem ) If $ \langle S_n \rangle$ , $ \langle T_n \rangle$ and $ \langle U_n \rangle$ be three sequences such that
- $ S_n \le T_n \le U_n, \ \forall n \in \mathbb{N}$
- $ \lim S_n=l= \lim U_n$ ,
then $ \lim T_n=l$ .
- ( Cauchy’s first theorem on Limits ) If $ \lim S_n =l$ , then $ \dfrac{1}{n} \{ S_1+S_2+ \ldots +S_n \} =l$ .
- ( Cauchy’s Second Theorem on Limits ) If $ \langle S_n \rangle$ is sequence such that $ S_n$ > $ 0, \ \forall n$ and $ \lim S_n =l$ , then $ \lim {(S_1 \cdot S_2 \cdot \ldots S_n)}^{1/n}= l$ .
- Suppose $ \langle S_n \rangle$ is a sequence of positive real numbers such that $ \lim \dfrac{S_{n+1}}{S-n} =l$ , ( l>0 ), then $ \lim {(S_n)}^{1/n}=l$ .
- ( Cesaro’s theorem ) If $ \lim S_n=l$ and $ \lim T_n=l’$ , then $ \lim \dfrac{1}{n} \{ S_1 T_1 + S_2 T_2 + \ldots + S_n t_n \} = l \cdot l’$
Theorem on Sub-Sequences
- If a sequence $ \langle S_n \rangle$ converges to l , then every subsequence of $ \langle S_n \rangle$ converges to l , i.e., every sub-sequence of a given sequence converges to the same limit.
Divergent Sequence
A sequence $ \langle S_n \rangle$ is said to diverge if $ \lim_{n \to \infty} S_n = +\infty$ or $ \lim_{n \to \infty} S_n = -\infty$ .
Oscillatory Sequence
- A sequence $ \langle S_n \rangle$ is said to oscillate finitely if
I. It’s bounded.
II. It neither converges nor diverges. - A sequence $ \langle S_n \rangle$ is said to oscillate infinitely, if
I. It’s not bounded.
II. It neither converges nor diverges.
A sequence is said to be non-convergent if it’s either divergent or oscillatory.
Limit Points of a Sequence
A real number P is said to be a limit point of a sequence if every neighborhood of P contains an infinite number of elements of the given sequence. In other words, a real number P is a limit point of a sequence $ \langle S_n \rangle$ , if for a given $ \epsilon$ >0, $ S_n \in (P-\epsilon, P+\epsilon )$ for infinitely many values of n .
Bolzano Weierstrass Theorem: Every bounded real sequence has a limit point. (Proof)
Remarks:
- An unbounded sequence may or may not have a limit point.
- The greatest limit point of the bounded sequence $ \langle S_n \rangle$ is called the limit superior of $ \langle S_n \rangle$ and is denoted by $ \lim \text{Sup} S_n$ .
- The smallest limit point of the bounded sequence $ \langle S_n \rangle$ is called the limit inferior of $ \langle S_n \rangle$ and is denoted by $ \lim \text{Inf} S_n$ .
- limSup $ \ge$ limInf.
Monotonic Sequences
A sequence $ \langle S_n \rangle$ is said to be monotonic if
either (i) $ S_{n+1} \ge S_n, \forall n \in \mathbb{N}$
or, (ii) $ S_{n+1} \le S_n, \forall n \in \mathbb{N}$ .
In first case, the sequence is said to be monotonically increasing while in the second case, it’s monotonically decreasing .
Important Theorems on Monotonic Sequences
- A monotonically increasing sequence, which is bounded above, is convergent. (Otherwise, it diverges to $ +\infty$ .) It converges to its supremum.
- A monotonically decreasing sequence, which is bounded below, is convergent. (Otherwise, it diverges to $ -\infty$ .) It converges to its infimum.
- A monotonic sequence is convergent iff it’s bounded. (<== combination of first two theorems ).
Cauchy Sequences
A sequence $ \langle S_n \rangle$ is said to be a Cauchy’s sequence if for every $ \epsilon$ >0, there exists a positive integer m such that $ |S_n -S_m|$ < $ \epsilon$ , whenever $ n \ge m$ .
Important Properties of Cauchy Sequences
- Every Cauchy sequence is bounded. (proof)
- ( Cauchy’s general principle of convergence ) A sequence of real numbers converges if and only if it is a Cauchy sequence. (proof)
$ \Box$