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Homework Help: Convergence of a series

  1. May 17, 2006 #1
    Dear all

    If a series e.g. a power series results in x convergering towards zero, can then one conclude that this series converge for all number if lets x belongs to R?

    Sincerely Yours
    Hummingbird25
     
  2. jcsd
  3. May 17, 2006 #2
    what do you mean by R? the ratio between terms?
     
  4. May 17, 2006 #3
    I believe Hummingbird was referring to the set of real numbers (often denoted by R). The answer to the question (if I read it correctly, I had to read it a few times) is no also, look up "radius of convergence".
     
  5. May 17, 2006 #4

    HallsofIvy

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    Science Advisor

    This makes no sense at all. "A power series results in x converging towards zero"? First of all, x does not "converge" toward anything. It is a variable. Second, I don't know what you mean by saying "a power series results" in that.

    If I really had to guess, I would guess you are asking about the "ratio test". If, for any series of positive numbers
    [tex]\Sum_{n=0}^\infnty a_n[/tex] the sequence [tex]\frac{a_{n+1}}{a_n}[/tex] converges to any number less than 1, then the series converges.

    From that it follows that if, for the power series [tex]\Sum_{n=0}^\infty a_nx^n[/itex] and some specific x, the ratio [tex]\left|\frac{a_{n+1}}{a_n}\right|\left|x\right|[/tex] is less than 1 then the series converges for that x. In particular, if [tex]\left|\frac{a_{n+1}}{a_n}\right|[/tex] converges to 0 then the above will converge to 0 <1 for all x and so the power series converges for all x.
     
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