Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (2.15)

\begin{equation*}
\underset{n\to \infty }{\mathop{\lim }}\,\,{{\beta }_{n}}\to \frac{\Gamma \left( 1-r \right)}{{{n}^{1-r}}}
\end{equation*} Proof:
\begin{align*}
\underset{n\to \infty }{\mathop{\lim }}\,\,{{\beta }_{n}}&=\underset{n\to \infty }{\mathop{\lim }}\,\,\frac{n!}{\left( 1-r \right)\left( 2-r \right)\cdots \left( n+1-r \right)} \\
&=\underset{n\to \infty }{\mathop{\lim }}\,\,\frac{1}{\left[ \left( 1-r \right) \right]\left[ 1+\frac{\left( 1-r \right)}{1} \right]\cdots \left[ 1+\frac{\left( 1-r \right)}{n} \right]} \\
&=\frac{1}{{{n}^{1-r}}}\left\{ \underset{n\to \infty }{\mathop{\lim }}\,\,\frac{{{n}^{1-r}}}{\left[ \left( 1-r \right) \right]\left[ 1+\frac{\left( 1-r \right)}{1} \right]\cdots \left[ 1+\frac{\left( 1-r \right)}{n} \right]} \right\} \\
&=\frac{\Gamma \left( 1-r \right)}{{{n}^{1-r}}}
\end{align*}
$\square$

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