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Calculus problem

  1. Jun 17, 2009 #1
    Suppose that
    [tex]
    \alpha_1,...,\alpha_n
    [/tex]
    are positive numbers. Show that
    [tex]
    \int_{1}^{\infty}...\int_{1}^{\infty}\frac{dx_1...dx_n}{{x_1}^{\alpha_1}+...+{x_n}^{\alpha_n}}<\infty
    [/tex]
    if
    [tex]
    \frac{1}{\alpha_1}+...+\frac{1}{\alpha_n}<1
    [/tex]
     
  2. jcsd
  3. Jun 18, 2009 #2
    Hi. I've been thinking about this one, but I cant solve it. Where did you get this problem?
     
  4. Jun 18, 2009 #3

    Mark44

    Staff: Mentor

    I would try breaking this down into smaller, easier problems. What do you get for this integral?
    [tex]
    \int_{1}^{\infty}\frac{dx_1}{{x_1}^{\alpha_1}+...+{x_n}^{\alpha_n}}
    [/tex]

    Note that this is an improper integral, so the limits will need to be 1 and, say, b, and you'll need to take the limit as b -> [itex]\infty[/itex].

    For the integral to converge, are there any restrictions on [itex]\alpha_1[/itex]?

    Then work with the double integral, with [itex]dx_1[/itex] and [itex]dx_2[/itex]. For this integral to converge, what restrictions must be placed on [itex]\alpha_1[/itex] and [itex]\alpha_2[/itex]?

    That's how I would tackle this.
     
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