  A recent asymptotic expansion for the positive zeros $x=j_{\nu,m}$
($m=1,2,3,\ldots$) of the Bessel function of the first kind $J_{\nu}(x)$ is
studied, where the order $\nu$ is positive. Unlike previous well-known
expansions in the literature, this is uniformly valid for one or both $m$ and
$\nu$ unbounded, namely $m=1,2,3,\ldots$ and $1 \leq \nu < \infty$. Explicit
and simple lower and upper error bounds are derived for the difference between
$j_{\nu,m}$ and the first three terms of the expansion. The bounds are sharp in
the sense they are close to the value of the fourth term of the expansion (i.e.
the first neglected term).
