Radius of convergence: 1/(1+x^2) about 1, using only real analysis

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Discussion Overview

The discussion centers on determining the radius of convergence for the function 1/(1+x^2) when expanded about the point x_0=1, with a focus on using real analysis methods. Participants express challenges in deriving the series and exploring related functions.

Discussion Character

  • Homework-related, Exploratory, Technical explanation

Main Points Raised

  • One participant references a previous thread for context and states that the expected radius of convergence is sqrt(2), but expresses difficulty in arriving at that conclusion.
  • Another participant inquires about the Taylor series and the tests that have been attempted, suggesting that it might be simpler to find the radius of convergence for 1/(1+x) about 1 instead.
  • A different participant notes the absence of an easy formula for the nth derivative of 1/(1+x^2), indicating uncertainty about how to proceed.
  • Another participant mentions having a nice form for the nth derivative of 1/(1+x), implying it could be relevant to the discussion.
  • A participant seeks clarification on whether the nth derivative form can be used to derive a Taylor series for 1/(1+x^2).

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are multiple approaches suggested and varying levels of understanding regarding the derivatives and series expansions.

Contextual Notes

There are limitations in the discussion regarding the lack of clear formulas for derivatives and the specific methods allowed under real analysis, which may affect the ability to derive the radius of convergence.

pcvt
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I've seen this thread:
https://www.physicsforums.com/showthread.php?t=297842

and that is the exact question I need to to answer.

What is the radius of convergence of 1/(1+x^2) expanded about x_0=1?

The problem is, I can only use an argument in real analysis.

I see the answer is sqrt(2), but I cannot get that answer.

I've tried a lot of little manipulations to get different geometric series but I can't seem to get it.

Thanks
 
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What is the taylor series? What tests did you try? It might be easier to find the radius of convergence of 1/(1+x) about 1.
 
I don't see an easy formula for the nth derivative so I'm not sure how to proceed in that direction
 
I didn't do 1/(1+x^2) but there is nice form of the nth derivative of 1/(1+x)
 
Do you mean I can use that to get a taylor series for my function
 

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