CONVERGENCE OF SEQUENCE AND SERIES (Part -1)| REAL ANALYSIS

INTRODUCTION:

All of you must read the 1st part of real analysis.If you are not then just CLICK HERE TO READ.

Though I repeat some basic knowledge about real analysis.

Real Sequence:
A real sequence is a map ∱: N ➝ R
If we write ∱(n)= x_n , n ∈ N 
then it is customary to denote the sequence
∱ by (x_n)_n=1 or sometimes (x₁ ,x₂ ,x₃ ,...,x_n )

The set {x_n : n ∈ N } is the range of the sequence.

A sequence in R is a list or ordered set: (a₁ , a₂ , a₃ , ... ) of real numbers.

Bounded Sequence:

The sequence  (x_n) is said to be bounded above if there exist a real number 'M' such that x_n ≤ M  for all n ∈ N .

       It is said to be bounded below if there exist m∈ R such that x_n ≥ m  for all n ∈ N.

A sequence is said to be bounded if it is bounded above and below.
So, simply (x_n) is bounded if and only if there exist a constant K > 0 such that

      |x_n| < K for n ∈ N
i.e;   -K < x_n < K

Convergence:

A sequence x_n  is called convergent if there exist a real number ' a ' such that for every є > 0 ,there exist n₀ ∈ N  depending upon є such that for n > n₀

                                         |x_n - a| < є

that is,

                               a - є < x_n < a+ є ,

Simply,

$$

Definition: We say x_n→∞ as n→∞ if for every M in R there is a natural number N so that x_n≥ M for all n≥N. We say x_n→−∞ as n→∞ if for every M in R there is a natural number N so that x_n≤ M for all n≥N.

Example: 1, 1/2, 1/4, 1/8, 1/16 ,.........

This Sequence converge to a a finite point so it is Convergence Sequence.

Divergence:

A sequence that is not converge is called Diverge.

Example: 1,2,3,4,......

In the above example the sequence leads to +∞ . So it is a Divergent Sequence.

Read the below Question Carefully for Better Understand.

QUESTION:

1. Check 1/2 , 2/3 , 3/4 ,4/5 ,5/6 ........   is Convergence or Divergence ?

ANSWER:

Step1: Find the n_th term

t_n = n/(n+1)

Step2: Find the limit of the n_th term with n↠ ∞

Similarly , if we don't get any finite number then it will be Divergent Sequence.

Limit Superior and Limit Interior Sequences:

Let x_n  be the sequence of real numbers , for any k, let S_k = { x_n : n > k} . Then

Also Limit Superior and Limit Interior is categories under Convergence and Divergence. Which we have already discussed above.

Other Types Of Sequences:

Monotonic Sequences:

A sequence of increasing(Diverging) or decreasing(Converging) is called Monotonic Sequences.

Example:

1,1/3,1/9 ,1/27 ,........ is a Converging Sequence.

0,2,4,6,8,........ is a Diverging Sequence.

Oscillatory Sequence:

The oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point.

Example: (-1) ⁿ  is an example of Oscillatory Sequence.

Cauchy's Sequence:

We say that a sequence of real numbers {a_n } is a Cauchy sequence provided that for every Є>0, there is a natural number N so that when n,m≥N, we have that|a_n -a_m |≤ Є.

Basic Properties of Cauchy's Sequence:

1. Every convergent sequence is a Cauchy sequence,
2. Every Cauchy sequence of real (or complex) numbers is bounded .
3. If in a metric space, a Cauchy sequence possessing a convergent sub sequence with limit is itself convergent and has the same limit.

BOLZANO-WEIERSTRASS THEOREM FOR SETS | REAL ANALYSIS

INTRODUCTION:

Have you remembered  Bounded set ,Order completeness of Real numbers.
If not just Click to Read Real Analysis part-1 .

Bounded Set:
A set is said to be bounded if it is bounded above and below.
" I will  not explain this just read about it in the above link "

Bolzano-Weierstrass Theorem:

Statement: Every infinite bounded set of real numbers has a limit point.

Proof: Let ' S ' be any infinite bounded set of real numbers.
                 
                  m ≤ x ≤ M ,      ∀  x ∈S  ---(1)
                     
    m = infimum of S       M=supremum of S

Let us assume T is a set of Real numbers.
 
          T = { x : x is greater than only for finite number of elements } ------(2)

 x ∈ T { x ≥ only a finite number of elements of S } --(3)

{ x ≤ infinite number of elements of S         }---(4)

  [ Don't confuse with 'S' and 'T' .   S is a infinitely bounded set while T is a finitely bounded set and T is the subset of S also element 'x' is common in both sets]

From equation (1) and (4)
               
                      m ∈ T  ⇔  T ≠ Φ

From equation (1)
               
                       M ≥ x  ,     ∀  x ∈S
 
Let us assume another element 'y' from set T.
                  y ∈ T  ⇔  y ≥ only a finite number of elements of S  [From equation (3)]

    M ≥ y ,    ∀  y ∈ T  ⇔ T is bounded above.

 Now we have prove that T is bounded above

So it will follow Order Completeness Property and we know that " There exist a supremum of  a set of which set follows the completeness property ".

       Let  P be the Supremum of T .
 
Claim: ( P  is a limit point  of set S )

          Let є > 0 be arbitrary number.

 Since   P + є > P
     
          ⇒ P + є ≥ finite number of elements of S   -----(5)
   [ Because P is the Supremum of T and T is the subset of S]
 
Since     P - є < P
      
         ⇒ P - є ≤  infinite number of elements of S ----(6)

From equation (5) and (6) that  nbh
      
                ］  P - є , P + є [   of  P contains infinite number of elements of S.

Hence , P is a limit point.


Example:

1.  Z  has no limit point .{it has not bounded}

    2.   S = { -1 ⁿ(n/(n+1)  : n ∈ N}

Ans. 
           S = { -1/2 ,2/3 , -3/4 .....}
          infinite | Bounded   -1 < x < 1

hence lmit point will  be 0.

If any doubt , Comment below

Dedekind's Form of Completeness | Real Analysis

Dedekind's Form of Completeness:

Definition:

If X and Y are two non-empty subsets of R such that
(i) X U Y = R
(ii) There is no common element between X and Y.
(iii) x < y { x€ X and y€ Y}

Summary:
If all the real numbers divided into two non-empty sets X and Y such that every element of X is less than every element of Y,Then there exist a unique real number says ' a ' such that every real number belongs to X is less than 'a' and every real number greater than 'a' is belongs to Y.

Clearly,the two classes X and Y are disjoint and the number 'a' itself belongs to either to X or Y. Then either X has GLB or Y has LUB(completeness).
This property of real numbers is called as Dedekind's Completeness Property.

Real Analysis (Chapter-1)

Analysis:
Definition:Mathematical analysis is the branch of mathematics which deals with limits and related theories such as differentiation,integration,infinite series and analytic Function. It is of two types Ex:(i)Real Analysis (ii)Complex Analysis
Real Analysis:
It is the study of sequence and their limits,continuity,differentiation,integration and sequence of function.It deals with the function of real variables.
Field:
A field is a set ' F ' with two binary operation denoted by ' + ' and ' . ' and must satisfy the following axioms.

[Properties] [Condition] Ex:[Addition] [Multiplication]

1.Closure	∀a,b∈F	a+b ∈F	a.b ∈F
2.Commutative	∀a,b∈F	a+b=b+a	a.b=b.a

3.Associative	∀a,b,c∈F	a+(b+c)=(a+b)+c	(a.b).c=a.(b.c)
4.Identity	∀a∈F	a+0=0+a=a	a.1=1.a=a

5.Inverse	∀a∈F	a+(-a)=0	a-1.a=0
6.Distributive	∀a,b,c∈F	a.(b+c)=a.b+a.c

Order:
Let  F be a field and let A,B∈F(here A and B are sets) then we define
                  A+B={x+y∈F : x ∈A and y∈B}

- A={-x : x∈A}
nA={nA: x ∈A} n∈N

In simple word :- A Field F is said to be ordered if there exist a subset P ⊂ F
such that
             (i)  P⋂(-P)=Φ
          (ii) P⋃{0}⋃(-P)=F
          (iii) a,b∈P  ⇒ a+b∈P and a.b∈P

[ i.e;P is a Field]

Bounded and Unbounded Set:
a set is called bounded if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded.

Upper Bound and Lower Bound of a Set:

A set M of real numbers is called bounded from above if there exists some real number k(not necessarily in M) such that k ≥ s for all s in M. The number k is called an upper bound of M. The terms bounded from below and lower bound are similarly defined.

A set S is bounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval.

[The 'Greatest Lower Bound' is called 'infimum ' and 'Least Upper Bound ' is called 'Supremum' .]

The Completeness Property:

The Completeness Property of The Real Numbers: Every nonempty subset S of the real numbers that is bounded above has a supremum in R.

The Completeness Property is also often called the "Least Upper Bound Property".

The Archimedean Property:

Archimedean property which tells us that for any real number x there exists a natural number n that is greater than or equal to x. 

    [Simply;  ∀ x∈R there exist an element 

n∈N such that  x ≤ n .]

Proof:(By Method of Contradiction)
Letus assume x,y€ R
And for all n€N ,nx<y (contradictory statements)
Let A={nx: x ∈A} n∈Nand y is the upper bound.
By Completeness axiom,there exist q = lub(A).There exist mx€N such that q-x < mx
So q < x(m+1)
Which is a contradiction.

The Density of the Rational/Irrational Numbers:

Density: A subset S of R is called dense in R if between any two real number there must be a element of S.

Let x,y∈R be any two real numbers where x<y. Then there exists a rational number r∈Qsuch that x<r<y.

Similarly ,Let x,y∈R be any two real numbers where x<y. Then there exists a irrational number q∈R/Q such that x<q<y.

Dedekind form of Completeness Property:

😊

We will discuss from here in next chapter.