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crybllrd
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Homework Statement
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
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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