I just learned induction in another thread and I'm curious if it can be used to prove that the gamma function converges for [itex]p\geq0[/itex]. I'm not sure if it can be used in this way. Is this wrong?(adsbygoogle = window.adsbygoogle || []).push({});

Gamma Function is defined as:

[tex]\Gamma(p+1)=\int_0^\infty e^{-x}x^p \,dx[/tex] We're trying to show that this converges for [itex]p\geq0[/itex]

Smallest case, p=0:

[tex]\Gamma(1)=1[/tex] converges

Assume the following converges:

[tex]\Gamma(p)=\int_0^\infty e^{-x}x^{p-1} \,dx[/tex]

Using integration by parts we find:

[tex]\Gamma(p+1)=p\Gamma(p)[/tex]

So since

[tex]\Gamma(p)[/tex] converges

then

[tex]\Gamma(p+1)=p\Gamma(p)[/tex] must also converge

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# Gamma Function Convergence

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