Absolutely convergent

  • Thread starter steven187
  • Start date
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hello all

well i think im kind of brain dead, iv been workin on alot of problems over the last few days, I cant 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...
 

Icebreaker

Try the squeeze theorem. What's with a_n, though? You mean each term has a different coefficient?
 

HallsofIvy

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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.
 

Icebreaker

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.
 

saltydog

Science Advisor
Homework Helper
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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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