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How to show $\Gamma (n)$ is convergent if and only $n>0$ where $\Gamma (n)$ is the gamma function.

esege
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  • Well a function value can't really be convergent... I suspect you are asking about the convergence of $\int_0^\infty t^{x-1}e^{-t},dt$ – user2345215 Apr 05 '14 at 14:35
  • Yes i'm asking that but i didn't understand the difference. – esege Apr 05 '14 at 14:42
  • The question's ill-posed. It seems to be you meant to ask to show the integral defining $;\Gamma(t);$ indeed converges iff $;\text{Re}(t)>0;$ . You must know the gamma function can be analitically continued to the whole complex plane without the non-positive integers... – DonAntonio Apr 05 '14 at 14:50

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Show that $\displaystyle\int_0^1t^{x-1}e^{-t}\,dt$ converges. Use the convergence of $\displaystyle\int_0^1t^{x-1}\,dt$ and the continuity of $e^{-t}$.

Show that $\displaystyle\int_1^\infty t^{x-1}e^{-t}\,dt$ converges. Use the convergence of $\displaystyle\int_1^\infty e^{-t/2}\,dt$ and that $\displaystyle\lim_{t\to\infty}\frac{t^{x-1}}{e^{t/2}}=0$.

user2345215
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