Homework Help: Finding Region of convergence for complex series

1. Apr 9, 2012

andrie55

1. The problem statement, all variables and given/known data

Find the region of convergence for the following complex series and draw the region

Ʃ(i+z)^(2n-1)/2^(2n+1)

3. The attempt at a solution

Here is my hand written working, sorry i could figure out how to use the symbols

http://i43.tinypic.com/10xsnrc.jpg

Thank you

I am sorry I just remembered that links to images are not permitted. How do I delete this thread?

Last edited: Apr 9, 2012
2. Apr 9, 2012

Dick

I think the figure is all right. Now you just want to figure out what region |(z+i)^2/4|<1 corresponds to. Simplify it a little.

3. Apr 9, 2012

andrie55

Thats the problem mate. I just don't have a clue on what to do after this stage. I have look a various text books and online without success.

4. Apr 9, 2012

micromass

You want to get it in the form $|z-c|<r$ for some c and r.

Hint:

1) |cz|=c|z| for positive real c
2) |z²|=|z|²

5. Apr 9, 2012

andrie55

To be honest I dont have a clue now as in most cases we have only worked out if the series in convergence or divergent. This is new to me. Are they any textbooks or websites u recommend for learning this stuff?

Anyway thanx for your help guys

6. Apr 9, 2012

Dick

You may be thinking this is much harder than it actually is. You've already done the hard part. Do you know |z+i|=|z-(-i)|<r is an open disk of radius r around the point z=(-i)?

7. Apr 10, 2012

andrie55

No I dont know that. So in this case c=-i?

8. Apr 10, 2012

Dick

Sure, |z-c|<r is a disk centered on c. So if c=(-i) then you get |z+i|<r. Now what's r?

9. Apr 10, 2012

andrie55

Since its convergence r is equals to 1. Unless i have to multiply by the denominator of 4?

10. Apr 10, 2012

Dick

Work it out. If |(z+i)^2/4|<1 then |(z-(-i)|<? No, it's not 1.

11. Apr 10, 2012

andrie55

cross multiplication makes it |(z+i)^2|<4
then square root of 4 =2
so |(z-(-i)|< 2

12. Apr 10, 2012

Dick

Yes. The radius of convergence is 2.

13. Apr 10, 2012

andrie55

Oh thank you very much I have learnt something very important. So now I just need to plot xy plane with i along the y axis and numbers (radius) along the x axis. The circle with initiate from point -i with radius of 2. So how will i know that the radius is 2 along the y axis since its a imaginary axis with no real numbers?

14. Apr 10, 2012

Dick

You are making this sound more complicated than it is. If y is your imaginary axis then the point -i is the point (0,-1). Use that as a center and draw a circle of radius 2 around it.