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

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The discussion focuses on the ratio test for proving absolute convergence of a power series, specifically the series \(\sum_{n=1}^{\infty} a_n x^n\). It is established that if the series converges for some \(x \neq 0\), then it is absolutely convergent for all \(w\) where \(|w| < |x|\). The ratio test is highlighted as a definitive method to determine absolute convergence, where the limit \(\mathop\lim\limits_{n\to\infty} |\frac{u_{n+1}}{u_n}|=L<1\) confirms convergence.

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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 x\not= 0, the series
\sum_{n=1}^{\infty} a_n x^n
is convergent. Prove the series is absolutely convergent for all w with |w|&lt;|x|.

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:

\mathop\lim\limits_{n\to\infty} |\frac{u_{n+1}}{u_n}|=L&lt;1

then the given series is absolutely convergent.
 

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