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Positive Radius of Convergence

  1. Feb 5, 2008 #1

    Suppose that {[tex]a_{k}[/tex]}[tex]^{\infty}_{k=0}[/tex] is a bounded sequence
    of real numbers. Show that [tex]\sum[/tex]a[tex]_{k}[/tex]x[tex]^{k}[/tex] has a
    positive radius of convergence.


    I have attempted to use the ratio test and failed. I am suspicious I can try the root
    test, but I am not sure how to work it. I just got used to 'Math Type Lite' and I am
    not used to Latex, hence it took me a while to type it up. Pardon if my question looks

    Anyway, I am depressed. I spent the past two hours on this problem and I am getting nowhere.

    Thank You,
  2. jcsd
  3. Feb 5, 2008 #2


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    Look at the hypotheses of the ratio test: you can use it only when you have a series whose terms are nonnegative. This could serve as a hint!
  4. Feb 5, 2008 #3
    I am thinking of using the root test. I know there is a theorem related to the root test that discusses R = lim sup [1/|a_k|^1/k ]. (Pardon my crappy lack of knowledge of Latex, I'm learning).

    So, what am I trying to say? I am not sure. I know that if a sequence is bounded, then
    the sequence should be less than some constant M, always. Yet, so what? How does
    this tell me that my R (radius of convergence) is positive?

    I hope this tells you where I have failed. Ugh, I really hate power series.

  5. Feb 5, 2008 #4


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    Suppose [itex]|a_k| <= M[/itex] for all k, where M>0. Then [itex]|a_k|^{1/k} \leq M^{1/k}[/itex] and thus [itex]\limsup_k |a_k|^{1/k} \leq \sup_k M^{1/k} < \infty[/itex].
  6. Feb 5, 2008 #5


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    Notice that all you really need to do is show that the series converges for some non-zero x. Then the radius of convergence must be larger than |x| and so is positive.

    You are told that the sequence {an} is bounded. That is, there exist some positive integer M such that |an|< M. What is the radius of convergence of
    [tex]\sum_{n=0}^{\infty} Mx^n[/itex]?
  7. Feb 5, 2008 #6

    Thank you all for your help. I think I got a solid result. I shored up the arguments used here with the root test. :-)

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