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Homework Help: When will the following converge/diverge?

  1. Apr 11, 2009 #1
    for what values of p, q will the following integral converge??

    intergal from (0 - ∞ ) of

    [tex]\int[/tex]dx/(x^p +x^q)

    i know that both [tex]\int[/tex]dx/(x^p) and [tex]\int[/tex]dx/(x^q) will converge when p,q>1 and i know that is smaller than either [tex]\int[/tex]dx/(x^p) or [tex]\int[/tex]dx/(x^q) alone since x>0, the denominator will be bigger so the fraction will be smaller.

    so p,q>1 converge

    but i dont think this is a proper answer, since as far as i see it is a possibility but not necessarily the only one,

    how do i reach the correct answer?
  2. jcsd
  3. Apr 11, 2009 #2


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    Science Advisor

    First of all, you need to be more precise; does "p,q>1" mean that both p and q are greater than one?

    Then, you seem to be thinking only about convergence as x approaches infinity; what about as x approaches zero?
  4. Apr 11, 2009 #3
    since i looked at dx/(x^p) and dx/(x^q) seperately i meant that they both were bigger than 1, but now looking at what i wrote and your answer, i am not sure that that wat the right thing to do, let me try explain.
    what i was looking at was the comarison where

    {g(x) > f(x)} and g(x) converges, then f(x) converges
    {g(x) > f(x)} and g(x) diverges, then f(x) diverges

    how would you go about solving this.
    i have another question based on the same principal, but a more complex function and would like to know this one before attempting it.
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