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Homework Help: Show that the Gamma function is converging

  1. Feb 23, 2012 #1
    1. The problem statement, all variables and given/known data

    The gamma function, which plays an important role in advanced applications, is defined for

    [itex]n\geq1[/itex] by [itex]\Gamma(n)=\int_0^{\infty} t^{n-1}e^{-t}dt[/itex]

    (a) Show that the integral converges on [itex]n\geq1[/itex]

    (b) Show that [itex]\Gamma(n+1)=n\Gamma(n)[/itex]

    (c) Show that [itex]\Gamma(n+1)=n! if n\geq1[/itex] is an integer

    3. The attempt at a solution

    I am only having trouble with part (a)

    (a) I used integration by parts to get

    [itex]e^{-t}nt^{n}+\int_0^{\infty} t^{n}e^{-t}dt[/itex]

    I also tried it with a different u and dv:

    [itex]-t^{n-1}e^{-t}+(n-1)\int_0^{\infty} t^{n-2}e^{-t}dt[/itex]

    I used this second IBP to find part (b), which I used for (c).

    How can I show it is converging?
     
    Last edited: Feb 23, 2012
  2. jcsd
  3. Feb 23, 2012 #2
    Never mind, I figured it out with a comparison test.
    Funny, after I post things on here I always seem to figure them out on my own.
    Well, whatever works!
     
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