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Math Help - Power series convergence

  1. #1
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    Power series convergence

    Prove that f(x) = Σ (k=0 to ∞) a_k x^k has a positive radius of convergence.

    Can someone give me some help on where to start with this?

    Thanks
    Last edited by taypez; December 16th 2007 at 05:33 PM. Reason: More info needed
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  2. #2
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    Quote Originally Posted by taypez View Post
    Prove that f(x) = Σ (k=0 to ∞) a_k x^k has a positive radius of convergence.

    Can someone give me some help on where to start with this?

    Thanks
    I really do not understand the question.

    The radius of convergence R is a number such that |x|<R the series converges absolutely. If |x|>R the series diverges. Now we say R = 0 if no such real number exists such that |x|<R implies convergence. And we say R = +oo if no such real number exists such that |x|>R implies divergence.
    So are you asking to prove that ever power series has: positive radius of convergence, zero radius of convergence, or infinite radius of convergence?
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  3. #3
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    power series

    Thanks. Unfortunately, the question was just as confusing to me. My thought was the same as yours that R had conditions and was not always positive. I thought maybe there was something that I was not understanding. I'm assuming now that I'm supposed to solve it for R>0. Who knows with this text!

    Thanks again.
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  4. #4
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    Prehaps we are given some facts about a_n?
    That would give the question some meaning.
    Is there more to the given?
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    That's the problem stated directly out of the book.
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  6. #6
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    Quote Originally Posted by taypez View Post
    That's the problem stated directly out of the book.
    What textbook are you using?

    Consider \sum\limits_n {n!x^n } this has a radius of convergence R=0.
    Here is a counter-example.
    Last edited by Plato; December 17th 2007 at 02:35 PM.
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