Converge of Complex Function

In summary, the conversation is about finding the set of values for z that will make the series, \sum (1/n!)(1/z)^n, absolutely convergent. The person asking for help is struggling with this type of problem and is looking for a logical approach to solving it. They are also grateful for the help they have received from others in the past.
  • #1
361
0
Hi,
In this Problem i am finding Problem to calculate the set of z:
Pls help
Determine all z [tex] \subset [/tex] C for which the following series is absolutely convergent:

[tex] \sum (1/n!)(1/z)^n [/tex]

Thx
 
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  • #2
Do the terms look familiar? What if you set [itex]w=\frac{1}{z}[/itex]?
 
  • #3
okay ...but generally how to solve such kind of problems..can u be more logical pls
 
  • #4
Use the ratio test (and i don't think you should accuse others of not being logical. pointing out that this is the series for exp{1/z} give or take a constant is very logical. you ought to put in the limits too).
 
  • #5
its just my concepts are not clear...how can i accuse anyone here..i have learned a lot from ppl here and i want to improve
Sometimes even in my class i end up asking such stupid questions that whole class bursts into laughter..
Thx for it and urs reply in pm...
 

1. What is the definition of "convergence" in the context of complex functions?

Convergence in the context of complex functions refers to the behavior of a sequence of complex numbers as the index of the sequence increases towards infinity. A sequence is said to converge if the terms in the sequence get closer and closer to a single limit value as the index increases.

2. How is the convergence of complex functions different from the convergence of real functions?

The convergence of complex functions differs from the convergence of real functions in that complex functions have two components, a real part and an imaginary part. This means that the behavior of a complex function is more complex and can exhibit different types of convergence, such as pointwise convergence and uniform convergence.

3. What are some common methods for proving the convergence of complex functions?

Some common methods for proving the convergence of complex functions include the Cauchy convergence test, the ratio test, and the root test. These tests involve analyzing the behavior of the sequence of complex numbers and determining if it approaches a limit or not.

4. Can a complex function converge to more than one limit?

No, a complex function can only converge to one limit. This is because a complex function can be represented as a single point in the complex plane, and a sequence of complex numbers can only approach one point in the plane as the index increases towards infinity.

5. How does the convergence of complex functions relate to the concept of continuity?

The convergence of complex functions is closely related to the concept of continuity. A complex function is said to be continuous if it is defined at all points in a given interval and its limit at each point exists and is equal to the value of the function at that point. Therefore, a function that has a limit at each point in an interval is also said to be convergent in that interval.

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