Let $$A = [a_{ij}] = \begin{bmatrix} 1& 1/2! & 1/3!\\ 1/2! & 1/3!& 1/4!\\ 1/3!& 1/4! & 1/5!\end{bmatrix}.$$
Let $A^{\circ r}=[a_{ij}^r]$ for $r>0$. We need to prove that $\det(A^{\circ r})$ is negative for every $r>1$. One observation is that $\det(A^{\circ r})$ is negative at $r = 1$, is increasing when $r>1$, and tends to zero as $r \rightarrow \infty$. This will serve the purpose.
Que: How can we show that $\det(A^{\circ r})$ is increasing when $r>1$?
The matrix $A$ is Hankel, is it helpful to prove this?
Any help or hint is appreciated.