How Does the Boundedness of Im(zn) Aid in Proving Convergence of <e^(i*zn)>?

In summary, the conversation is focused on proving that e^(i*zn) has a convergent subsequence when <zn> is a sequence of complex numbers with bounded Im(zn) values. The B-W theorem is mentioned as a possible help, but the sequence in question is only bounded below and on the imaginary part. The concept of compactness is also brought up. The conversation continues with discussing the behavior of e^z in the complex plane and how it forms a circle. The suggestion is made to show that the sequence is bounded, possibly using the inequality |e^{iz_n}| <= C. The speaker also mentions that they see the sequence as a circle and asks for clarification on what attribute they may be missing.
  • #1
gestalt
6
0
Let <zn> be a sequence complex numbers for which Im(zn) is bounded below.
Prove <e^(i*zn)> has a convergent subsequence.

My question on this is what possible help could the boundedness of the Im(zn) to this proof and what theorem might be of help?
 
Physics news on Phys.org
  • #2
Do you know anything about compactness or Bolzano-Weierstrass??
 
  • #3
do you know how e^z behaves geometrically?
 
  • #4
Yes, I do know the B-W theorem. I looked at it but it applies to bounded sequences. This sequence is only bounded below and only on the I am part. I suppose since you asked you see a way it is still applicable.

It was my understanding e^z forms a circle in the complex. Is this what you mean?
 
Last edited:
  • #5
gestalt said:
Yes, I do know the B-W theorem. I looked at it but it applies to bounded sequences. This sequence is only bounded below and only on the I am part. I suppose since you asked you see a way it is still applicable.

It was my understanding e^z forms a circle in the complex. I this what you mean?

Try to show that the sequence is bounded. Try to show that there is a constant C such that

[tex]|e^{iz_n}|\leq C[/tex]
 
  • #6
Edit: The latex was not loading but now I see what you mean. I apologize if this sounds ignorant but I did not think e^(i*zn) was bounded. If you are saying I should show it is then I assume it can be.
When I think of this sequence I must be seeing it all wrong. could you give any explanation of how it behaves. If it helps I see nothing but a circle when I think of it. What attribute am I missing?
 
Last edited:
  • #7
circles are bounded, right?
 

FAQ: How Does the Boundedness of Im(zn) Aid in Proving Convergence of <e^(i*zn)>?

What is the definition of a sequence of complex numbers?

A sequence of complex numbers is a list of numbers in which each term is associated with a unique positive integer. The terms in the sequence can be real or complex numbers.

How is a sequence of complex numbers represented?

A sequence of complex numbers is usually represented using the notation {zn}, where n is the positive integer index of the term. For example, z1, z2, z3, ... represents a sequence of complex numbers.

What is the difference between a finite and infinite sequence of complex numbers?

A finite sequence of complex numbers has a specific number of terms, while an infinite sequence continues indefinitely. In other words, a finite sequence has a last term, while an infinite sequence does not.

What is the limit of a sequence of complex numbers?

The limit of a sequence of complex numbers, if it exists, is the value that the terms of the sequence approach as the index n becomes very large. In other words, it is the value that the terms of the sequence get closer and closer to as n gets bigger.

How is the convergence of a sequence of complex numbers determined?

The convergence of a sequence of complex numbers is determined by taking the limit of the sequence and checking if it exists. If the limit exists, the sequence is said to be convergent. If the limit does not exist, the sequence is said to be divergent.

Similar threads

B Understanding about Sequences and Series
Replies
3
Views
1K
I Question about uniform convergence in a proof
Replies
1
Views
1K
A Question about proof of Great Picard Theorem
Replies
3
Views
1K
I Convergence of a recursively defined sequence
Replies
18
Views
3K
Physical details of the operation of galvanic cells
Replies
2
Views
2K
B How can hyperreal numbers make infinitesimals logically sound in calculus?
Replies
24
Views
4K
I Understanding the nth-term test for Divergence
Replies
17
Views
3K
Proving convergence of sequence from convergent subsequences
Replies
15
Views
1K
B Proving the Existence and Value of a Derivative at a Point
Replies
11
Views
1K
I What is the difference between these two power series theorems?
Replies
5
Views
920

Hot Threads

Recent Insights

Back
Top