1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook