Science Magazine

Real Sequences

Posted on the 12 August 2013 by Gaurav Tiwari @wpgaurav

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

  1. 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
.

  1. \langle {(-1)}^n \rangle
    is the sequence
    \langle -1, 1, -1, 1, \ldots, {(-1)}^n, \ldots \rangle
    .
  2. \langle -3n \rangle
    is the sequence
    \langle -3, -6, -9, \ldots, -3n, \ldots \rangle
  3. 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

  1. (Uniqueness Theorem) Every convergent sequence has a unique limit.
  2. For a sequence
    \langle S_n \rangle
    of non-negative numbers,
    \lim S_n \ge 0
    .
  3. Every convergent sequence is bounded, but the converse is not necessarily true.
  4. 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'
    .
  5. 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
    .
  6. 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|
    .
  7. (Sandwitch Theorem) If
    \langle S_n \rangle
    ,
    \langle T_n \rangle
    and
    \langle U_n \rangle
    be three sequences such that
    1. S_n \le T_n \le U_n, \ \forall n \in \mathbb{N}
    2. \lim S_n=l= \lim U_n
      ,
      then
      \lim T_n=l
      .
  8. (Cauchy’s first theorem on Limits) If
    \lim S_n =l
    , then
    \dfrac{1}{n} \{ S_1+S_2+ \ldots +S_n \} =l
    .
  9. (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
    .
  10. 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
    .
  11. (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

  1. 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


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