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Tricky series radius of convergence question (analysis course)

  1. Nov 21, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the radius of convergence of sum from 1 to n of

    1/(n^n) * x^(2^n)

    2. Relevant equations

    3. The attempt at a solution
    Clearly ratio test isn't going to work straight away. I'm not sure how to deal with the 2^n exponent
  2. jcsd
  3. Nov 21, 2011 #2


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    What about the nth root test?
  4. Nov 23, 2011 #3
    From there I get down to finding for what values the limsup of 1/n x^(2^n/n) is less than one for

    I believe its |x|<=1 but I'm not sure how to analytically prove this
  5. Nov 23, 2011 #4


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    I don't think |x|=1 works. But the case |x|<1 is easy. And |x|>1 is not much harder, you just want to show it diverges somehow. I'd say the easiest way is to combine l'Hopital or a ratio test with a comparison test. For example, can you show 2^n/n>n for large enough values of n?
    Last edited: Nov 23, 2011
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