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

Welcome all math lovers!