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Proving convergence given inequalities of powers

  1. Mar 27, 2014 #1
    1. The problem statement, all variables and given/known data

    Show that if a>-1 and b>a+1 then the following integral is convergent:

    ∫(x^a)/(1+x^b) from 0 to ∞



    3. The attempt at a solution

    x^-1 < x^a < x^a+1 < x^b

    x^-1/(1+x^b) < x^a/(1+x^b) < x^a+1/(1+x^b) < x^b/(1+x^b)

    I also know any integral of the form ∫1/x^p when p>1 is convergent (from any number t to ∞)

    Honestly not sure how to attack this problem. I'm trying to bound it but not sure how to show the parameters.
     
  2. jcsd
  3. Mar 28, 2014 #2

    pasmith

    User Avatar
    Homework Helper

    You need to show that both
    [tex]
    \lim_{\epsilon \to 0^{+}}\int_\epsilon^1 x^a(1 + x^b)^{-1}\,dx
    [/tex]
    and
    [tex]
    \lim_{R \to \infty} \int_1^R x^a(1 + x^b)^{-1}\,dx
    [/tex]
    converge. Since [itex]b - 1 > a > -1[/itex] we have [itex]b > 0[/itex], so [itex]0 < x^b < 1[/itex] if [itex]0 < x < 1[/itex] and [itex]0 < x^{-b} < 1[/itex] if [itex]x > 1[/itex] so you can use the binomial theorem to expand [itex](1 + x^b)^{-1}[/itex] in powers of [itex]x^b[/itex] or [itex]x^{-b}[/itex] as appropriate.
     
  4. Mar 29, 2014 #3
    Thank you!
     
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