Is There a Power Series That Converges at One Point and Diverges at Another?

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Homework Help Overview

The discussion revolves around the existence of a power series that converges at a specific complex point and diverges at another complex point. The subject area involves complex analysis and the properties of power series, particularly focusing on convergence criteria.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are exploring the conditions under which a power series can converge or diverge at given complex points. Questions about the center of the power series and the concept of radius of convergence are raised.

Discussion Status

The discussion is ongoing, with participants clarifying the original question and considering relevant theorems related to power series convergence. There is an exploration of different interpretations regarding the form of the power series and its implications for convergence.

Contextual Notes

There is some ambiguity regarding the specific points mentioned and whether the power series is centered at a particular location, which may affect the analysis of convergence.

mariab89
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Homework Statement



does there exist a power series that converges at z= 2+31 and diverges at z=3-i

Im really stuck on this one! any ideas?
 
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Do you mean 2+3i and 3-i? And do you mean a power series centered at z=0? There is a theorem about convergence of power series based on a radius of convergence. Can you find it?
 
Yes, sorry my question is to determine whether there exists a power series that converges at z = 2 + 3i and diverges at z = 3 - i.
 
Ok, so is the power series just a sum of z^i (as opposed to (z-c)^i)? And what do you know about 'radius of convergence'?
 

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