What is the ratio test for proving absolute convergence of a series?

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Discussion Overview

The discussion revolves around the ratio test for proving the absolute convergence of a power series. Participants explore the conditions under which a series converges absolutely, particularly in relation to its radius of convergence.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant, Steven, presents a problem regarding the absolute convergence of the series \(\sum_{n=1}^{\infty} a_n x^n\) for values of \(w\) where \(|w| < |x|\).
  • Another participant notes that a power series diverges outside its radius of convergence and converges absolutely within it.
  • There is a suggestion to apply the squeeze theorem, with questions raised about the coefficients \(a_n\) in the series.
  • Some participants express differing views on whether the coefficients must be the same to find the sum of the series.
  • A later reply introduces the ratio test, stating that if \(\mathop\lim\limits_{n\to\infty} |\frac{u_{n+1}}{u_n}|=L<1\), then the series is absolutely convergent.

Areas of Agreement / Disagreement

Participants express differing opinions on the nature of the coefficients in the series and the application of the ratio test. There is no consensus on the necessity of having the same coefficient for convergence or the best approach to proving absolute convergence.

Contextual Notes

Some assumptions about the coefficients \(a_n\) and their implications for convergence remain unresolved. The discussion also reflects varying interpretations of the ratio test's application.

steven187
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hello all

well i think I am kind of brain dead, iv been workin on a lot of problems over the last few days, I can't see anything obvious anymore, well this shall be the last one for today (i hope), anyway here it is,

suppose that for some [tex]x\not= 0[/tex], the series
[tex]\sum_{n=1}^{\infty} a_n x^n[/tex]
is convergent. Prove the series is absolutely convergent for all [tex]w[/tex] with [tex]|w|<|x|[/tex].

Steven
 
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Well, a power series diverges outside its radius of convergence and converges absolutely on the inside...
 
Try the squeeze theorem. What's with a_n, though? You mean each term has a different coefficient?
 
Icebreaker said:
Try the squeeze theorem. What's with a_n, though? You mean each term has a different coefficient?

Well, yes! That is the basic idea of a power series after all.
 
Odd, I have the idea in my head that they must have the same coefficient in order to find its sum, if it's convergent.
 
Hello Steven.

How about using the ratio test:

If:

[tex]\mathop\lim\limits_{n\to\infty} |\frac{u_{n+1}}{u_n}|=L<1[/tex]

then the given series is absolutely convergent.
 

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