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What is the radius of convergence of

  1. Feb 9, 2015 #1
    1. The problem statement, all variables and given/known data

    z ∈ ℂ

    What is the radius of convergence of (n=0 to ∞) Σ anzn?

    2. Relevant equations

    I used the Cauchy-Hardamard Theorem and found the lim sup of the convergent subsequences.

    [tex]a_n = \frac{n+(-1)^n}{n^2}[/tex]

    limn→∞ |an|1/n

    3. The attempt at a solution

    I think that the radius of convergence is one, i.e. |z| < 1. I figured that the numerator would tend to n with the oscillating 1 and so you'd get [tex]\frac{n^{1/n}}{n^{2/n}} = 1[/tex]

    R = 1/limn→∞ |an|1/n
     
  2. jcsd
  3. Feb 9, 2015 #2

    Dick

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    That's a little informal but it looks fine. You might want worry about what happens at z=1 and z=(-1) if you are concerned about the boundary cases.
     
  4. Feb 10, 2015 #3
    I assume a singularity is there and outside of the disk the series is divergent. Well, I'm leaving out a few bits of information in my "proof" here. I'll state a bit more of the background. What would a more formal proof look like?
     
  5. Feb 10, 2015 #4

    Dick

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    Rather than just ignoring the (-1)^n handle it with a squeeze type thing. E.g. n/2<=n+(-1)^n<=2n. You can easily find the outer limits. Discuss what happens when z=1 or z=(-1).
     
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