  It is a well-known fact that multivariate Student $t$ distributions converge
to multivariate Gaussian distributions as the number of degrees of freedom
$\nu$ tends to infinity, irrespective of the dimension $k\geq1$. In particular,
the Student's value at the mode (that is, the normalizing constant obtained by
evaluating the density at the center)
$c_{\nu,k}=\frac{\Gamma(\frac{\nu+k}{2})}{(\pi \nu)^{k/2} \Gamma(
\frac{\nu}{2})}$ converges towards the Gaussian value at the mode
$c_k=\frac{1}{(2\pi)^{k/2}}$. In this note, we prove a curious fact:
$c_{\nu,k}$ tends monotonically to $c_k$ for each $k$, but the monotonicity
changes from increasing in dimension $k=1$ to decreasing in dimensions $k\geq3$
whilst being constant in dimension $k=2$. A brief discussion raises the
question whether this \emph{a priori} curious finding is a curiosity, \emph{in
fine}.
