When will the following converge/diverge?

  • Thread starter Dell
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  • #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 don't 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
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?
  • #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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