Indefinite Integration of the Gamma Integral
and Related Statistical Applications

Proof of Equation (2.17)

\begin{equation*}
\underset{n\to \infty }{\mathop{\lim }}\,\,\sum\limits_{i=1}^{n}{{{w}_{i}}{{\beta }_{i}}}\to \infty
\end{equation*} Proof:
\begin{align*}
\underset{n\to \infty }{\mathop{\lim }}\,\,\sum\limits_{i=1}^{n}{{{w}_{i}}{{\beta }_{i}}}&>\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{i=1}^{n}{\left( 1\cdot {{\beta }_{n}} \right)\,} \\
& =\underset{n\to \infty }{\mathop{\lim }}\,\left( n\cdot {{\beta }_{n}} \right) \\
& =\underset{n\to \infty }{\mathop{\lim }}\,\frac{n\Gamma \left( 1-r \right)}{{{n}^{1-r}}} \\
& =\Gamma \left( 1-r \right)\underset{n\to \infty }{\mathop{\lim }}\,\left( {{n}^{r}} \right) \\
& \to \infty
\end{align*}
$\square$

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