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)= xn , n ∈ N
then it is customary to denote the sequence
∱ by (xn)n=1 or sometimes (x1 ,x2 ,x3 ,...,xn )
The set {xn : 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 (xn) is said to be bounded above if there exist a real number 'M' such that xn ≤ M for all n ∈ N .
It is said to be bounded below if there exist m∈ R such that xn ≥ m for all n ∈ N.
A sequence is said to be bounded if it is bounded above and below.
So, simply (xn) is bounded if and only if there exist a constant K > 0 such that
|xn| < K for n ∈ N
i.e; -K < xn < K
The sequence (xn) is said to be bounded above if there exist a real number 'M' such that xn ≤ M for all n ∈ N .
It is said to be bounded below if there exist m∈ R such that xn ≥ m for all n ∈ N.
A sequence is said to be bounded if it is bounded above and below.
So, simply (xn) is bounded if and only if there exist a constant K > 0 such that
|xn| < K for n ∈ N
i.e; -K < xn < K
Convergence:
A sequence xn 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
|xn - a| < є
that is,
a - є < xn < a+ є ,
Simply,
Definition: We say xn→∞ as n→∞ if for every M in R there is a natural number N so that xn≥ M for all n≥N. We say xn→−∞ as n→∞ if for every M in R there is a natural number N so that xn≤ 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 nth term
tn = n/(n+1)
Step2: Find the limit of the nth 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 xn be the sequence of real numbers , for any k, let Sk = { xn : 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 {an } 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|an -am |≤ Є.
Basic Properties of Cauchy's Sequence:
- Every convergent sequence is a Cauchy sequence,
- Every Cauchy sequence of real (or complex) numbers is bounded .
- If in a metric space, a Cauchy sequence possessing a convergent sub sequence with limit is itself convergent and has the same limit.
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