Mathematics - Limits Question with Solution | TestHub

Taking logarithms and rearranging the factorial product,

$$\text{log}\beta =\underset{n\to \infty }{\lim }\left[\frac{1}{n^2}\sum _{r=1}^n(n+1-r)\text{log}r-\alpha \text{log}n\right].$$

Writing $\text{log}r=\text{log}n+\text{log}(r/n)$ gives

$$\text{log}\beta =\underset{n\to \infty }{\lim }\left[\left(\frac{n+1}{2n}-\alpha \right)\text{log}n+\frac{1}{n}\sum _{r=1}^n\left(1+\frac{1}{n}-\frac{r}{n}\right)\text{log}\frac{r}{n}\right].$$

A finite, non-zero limit requires $\alpha =\frac{1}{2}$. The remaining Riemann sum tends to ${\int }_0^1(1-t)\text{log}tdt=-\frac{3}{4}$, so $\beta =e^{-3/4}$.