Taylor series radius of convergence and center

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SUMMARY

The discussion focuses on the Taylor series, specifically its radius of convergence and the implications of its center point. A Taylor series centered at a point 'a' converges within the interval [a-R, a+R]. It is established that there exists a Taylor series expansion centered at any point within this interval, although the behavior at the endpoints remains uncertain. The conversation also addresses how to determine the existence of a Taylor series expansion at a different point 'b' within the interval, emphasizing the need to analyze the series centered at 'a'.

PREREQUISITES
  • Understanding of Taylor series and their properties
  • Knowledge of convergence intervals in mathematical analysis
  • Familiarity with the concept of series expansion
  • Basic calculus skills, particularly limits and continuity
NEXT STEPS
  • Study the concept of Taylor series and their convergence criteria
  • Explore the relationship between radius of convergence and endpoints
  • Learn about analytic functions and their Taylor series representations
  • Investigate methods for determining the center of convergence for Taylor series
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Mathematicians, students in calculus or analysis courses, and anyone interested in understanding the properties of Taylor series and their applications in approximating functions.

azay
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When approximating a function with a Taylor series, I understand a series is centered around a given point a, and converges within a certain radius R. Say for a series with center a the interval of convergence is [a-R, a+R].

Does this imply that:

1. There also exists a Taylor series expansion centered in any of the other points in that interval?

2. If not, and one would like to describe the function in terms of Taylor series, in some interval [b-S, b+S] of a point b \in [a-R, a+R] and b \neq a. And let this interval [b-S, b+S] \subset [a-R, a+R]. If there exists no Taylor series expansion centered in b, it is still possible to say something about the interval [b-S, b+S] by looking at the expansion centered in a. But how could one possibly know that a series then does exist in a? How do you find this a?
 
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azay said:
1. There also exists a Taylor series expansion centered in any of the other points in that interval?

Yes, centered at any other point of the interval (a-R,a+R) . We don't know about the endpoints.
 

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