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Domain of a function

  1. Nov 15, 2009 #1
    1. The problem statement, all variables and given/known data

    What is the domain of the following function?

    [tex] f(x,y) = (\sqrt(x) + x\sqrt(y)) \sum_{k=1}^{\infinity} (k^3 + k) x^k y^k [/tex]
    when [itex] D_f \subset \Re^2 [/itex].

    3. The attempt at a solution

    The domain is [itex]x>0 \in \Re[/itex] and [itex]y>0 \in \Re[/itex] in my opinion.
     
    Last edited: Nov 15, 2009
  2. jcsd
  3. Nov 15, 2009 #2

    Mark44

    Staff: Mentor

    If f(x, y) were defined only as [itex]\sqrt{x} + x\sqrt{y}[/itex], that would be correct. However, I think you need to take the summation into account, and determine for which x and y the series converges.
     
  4. Nov 15, 2009 #3
    The summation is to infinity.
    The sum is always positive so the range is always real.
    This would suggests me that x > 0 and y > 0.

    However, I am uncertain, since the solution should not be that easy.
     
  5. Nov 15, 2009 #4

    Mark44

    Staff: Mentor

    The sum is probably convergent only for x and y in some disk centered at the origin. Outside that disk the sum would be divergent, hence the function would be undefined. It really seems to me that if you want to find the domain for this function, you need to see which values of x and y are such that the series converges.
     
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