Complex Analysis Radius of Convergence.

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  • #1
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Hello, I have two questions regarding the Radius of convergence.

1. What should we do at the interval (R-eps, R)
2. It used definition to prove radius of convergence, but I am not sure if it is necessary-sufficient condition of convergence. I get that this can be a sufficient condition but not sure of the necessity
 

Attachments

  • 스크린샷 2016-06-21 오후 4.01.05.png
    스크린샷 2016-06-21 오후 4.01.05.png
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  • RadiusOfConv.png
    RadiusOfConv.png
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Answers and Replies

  • #2
FactChecker
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If I understand the question, this may help -- The ε was picked arbitrarily small so the proof shows convergence all the way to ε = 0. So there really is no specific interval (R-ε, R) to worry about.
 
  • #3
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what I am concerend about is,



hm the author used (L+eps)|Z|<1 to argue that L^n*|Z|^n < (L+eps)^n|Z|^n < r^n whereas r<1

but the thing is as I showed at the uploaded png file (if the derivation is correct), it is clear that |Z|= R-eps satisfies the condition |Z| < R and if |Z|= R-eps, r = 1.

which implies (L+eps)|Z|<1 cannot be true. for all |Z|<R
Ah, also I made a mistake. It should be R+R^2*eps-eps-R*eps^2



hmm do you agree?
 
Last edited:
  • #4
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If I understand the question, this may help -- The ε was picked arbitrarily small so the proof shows convergence all the way to ε = 0. So there really is no specific interval (R-ε, R) to worry about.


maybe I screwed up at somewhere.
 

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    RadiusOfConvergence.png
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  • #5
FactChecker
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maybe I screwed up at somewhere.
Maybe not. But the point is that ε >0 is arbitrarily small (or r is arbitrarily near 1) so the the interval you are worried about shrinks down to nothing.
 
  • #6
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Maybe not. But the point is that ε >0 is arbitrarily small (or r is arbitrarily near 1) so the the interval you are worried about shrinks down to nothing.

Although we ignore the interval, the argument that (L+eps)|Z|<1 does not hold true if we take |Z| = R - eps. (I guess) It just bugs me and I am stuck at that page ;(

and my last conclusion R<1 doesn't seem right at all too.
 
  • #7
Stephen Tashi
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It isn't clear to me what is to be proven. We've only been shown the last part of the text that states "the theorem".
 
  • #8
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It isn't clear to me what is to be proven. We've only been shown the last part of the text that states "the theorem".

I am sorry.
 

Attachments

  • 스크린샷 2016-06-23 오후 1.03.35.png
    스크린샷 2016-06-23 오후 1.03.35.png
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