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Absolutely convergent

  1. Jun 19, 2005 #1
    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].

  2. jcsd
  3. Jun 19, 2005 #2
    Well, a power series diverges outside its radius of convergence and converges absolutely on the inside...
  4. Jun 19, 2005 #3
    Try the squeeze theorem. What's with a_n, though? You mean each term has a different coefficient?
  5. Jun 19, 2005 #4


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    Staff Emeritus
    Science Advisor

    Well, yes! That is the basic idea of a power series after all.
  6. Jun 19, 2005 #5
    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.
  7. Jun 19, 2005 #6


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    Homework Helper

    Hello Steven.

    How about using the ratio test:


    [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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