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Analysis: Absolute convergence, rearranging terms, power series question

  1. Nov 23, 2011 #1
    1. The problem statement, all variables and given/known data
    I can't figure out what my professor means in the last two lines:

    10zcns9.png

    I'm trying to prove why the product of two analytic functions is analytic, and I think that I am going to need to use a similar construction.

    3. The attempt at a solution

    So far to prove the product is analytic, I have said that for
    f(x) convergent for x<R1
    g(x) convergent for x<R2
    Then, (f*g)(x) is convergent for x<min(R1,R2), then, I think I can say that for some rearrnagment of terms, (f*g)(x) = sum n=0 to infinity of (c_n x^n),

    but I'm not sure if or why I could make that last step.

    Thanks!
     
  2. jcsd
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