(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Show that the Gamma function is converging

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