Radius on convergence in the complex plane

  • Thread starter racland
  • Start date
6
0

Main Question or Discussion Point

Anyone knows anything about the Radius on convergence in the complex plane (Complex Analysis)
 

Answers and Replies

452
0
Yes. Somebody knows something about the radius of convergence in the complex plane.

Hope I was helpful
 
ranger
Gold Member
1,654
1
Yes. Somebody knows something about the radius of convergence in the complex plane.

Hope I was helpful
:rofl: :rofl:
 
HallsofIvy
Science Advisor
Homework Helper
41,734
893
Radius of convergence in the complex plane is exactly the same as radius of convergence in the real numbers. Exactly what is your question?
 
6
0
Great

The problem states:
Find the Radius of Convergence of the following Power Series:
(a) Sumation as n goes from zero to infinity of Z^n!
(b) Sumation as N goes from zero to infinity of (n + 2^n)Z^n

For (a) I think the radius of convergence is 1 but I'm a bit unsure of that...
 
HallsofIvy
Science Advisor
Homework Helper
41,734
893
As I said- same thing as on the real line (except now it really is a radius!). Apply the "ratio" test: a series [itex]\Sigma a_n[/itex] converges absolutely if the limit ration
[tex]\lim_{n\rightarrow \infty} \left|\frac{a_{n+1}}{a_n}\right|[/tex]
is less than 1.
(Diverges if that limit is greater than one, may converge absolutely or conditionally or diverge if it is equal to 1).

In particular, for Zn, we have |Zn+1/Zn|= |Z|. That series converges absolutely for |Z|< 1. (It obviously diverges for Z= 1 or -1 and diverges for |Z|> 1)

Try (n+ 2n)Zn yourself.
 
StatusX
Homework Helper
2,563
1
Is that [itex]z^{n!}[/itex]? If so, you can prove a general result that if a_n is any increasing sequence of natural numbers, then [itex]z^{a_n}[/itex] converges iff |z|<1. This is the case HallsofIvy did if a_n=n, and yours if a_n=n!. The general proof follows from the result for a_n=n (which is the smallest increasing sequence of natural numbers).
 
Last edited:

Related Threads for: Radius on convergence in the complex plane

Replies
3
Views
5K
  • Last Post
Replies
5
Views
3K
Replies
1
Views
1K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
9
Views
2K
  • Last Post
Replies
4
Views
4K
Replies
1
Views
2K
Top