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Convergence of power series

  1. May 13, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose sum(a_n*x^n) represents a power series with radius of convergence (-R, R). Is it true that the series sum(n*a_n*x^n) is convergent? Prove or give a counter example.

    2. Relevant equations

    3. The attempt at a solution

    Let b_n = n*a_n*x^n
    Using ratio test:
    lim[b_(n+1)/b_n] = lim [(n+1)/n] * [lim x*a_(n+1)/a_n] < 1 because lim x*a_(n+1)/a_n <1 from hypothesis.

    Any gaps in logic?
  2. jcsd
  3. May 13, 2008 #2


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    I think that's correct.
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