Proving Convergence of Power Series for All x within Radius of Convergence

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Homework Help Overview

The discussion revolves around proving the convergence of a power series at all points within its radius of convergence, specifically when it converges at a certain point d. The original poster seeks to understand the implications of convergence at point d for other points x within the interval defined by the radius of convergence.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to clarify whether they should demonstrate that all x satisfying |x-x0|<|d-x0| also leads to convergence, and questions if they are expected to prove a general rule regarding power series convergence.
  • Some participants suggest not to assume the existence of the radius of convergence and to consider first principles in their approach.
  • There are inquiries about the role of ε in relation to the radius of convergence and how it should be applied in the proof.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and offering guidance on how to approach the proof. There is an emphasis on revisiting foundational concepts and clarifying assumptions without reaching a consensus on the specific path forward.

Contextual Notes

Participants are navigating the implications of convergence criteria and the definitions surrounding the radius of convergence, with some uncertainty about the assumptions that can be made in the proof.

peripatein
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Hi,

Homework Statement


I am asked to prove that if the power series Ʃ(1,infinity) a_n(x-x0)^n converges at a point d, then it converges for every x that satisfies |x-x0|<|d-x0|.


Homework Equations





The Attempt at a Solution


Obviously |d-x0|<r, where r denotes the radius of convergence. Should I strive to demonstrate that every such x would also satisfy |x-x0|<r? Is that the way to go about proving the above proposition? There is a general rule that if a power series converges at a point a, then it converges at every point x satisfying |x|<|a|. Should I therefore strive to show that |x|<|d|, or am I actually requested to prove the general rule? I'd appreciate some guidance.
 
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It seems to me you are being asked to prove that there is such a thing as radius of convergence, so better not to assume it. But it is hard to be sure.
 
What would you suggest then?
 
If in doubt, go back to first principles. Given ε > 0, there exists... You should be able to use this criterion wrt x-x0 to show satisfied for some x closer x0.
 
So what keeps me from letting epsilon be the radius itself?
 
peripatein said:
So what keeps me from letting epsilon be the radius itself?
ε would be a bound in relation to the value to which the series converges, not in relation to x. Btw I should have said 'use the fact that this criterion is satisfied wrt d-x0 ...'.
 

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