  We examine convergent representations for the sum of a decaying exponential
and a Bessel function in the form \[\sum_{n=1}^\infty
\frac{e^{-an}}{(\frac{1}{2} bn)^\nu}\,J_\nu(bn),\] where $J_\nu(x)$ is the
Bessel function of the first kind of order $\nu>-1/2$ and $a$, $b$ are positive
parameters. By means of a double Mellin-Barnes integral representation we
obtain a convergent asymptotic expansion that enables the evaluation of this
sum in the limit $a\to 0$ with $b<2\pi$ fixed. A similar result is derived for
the sum when the Bessel function is replaced by the modified Bessel function
$K_\nu(x)$. The alternating versions of these sums are also mentioned.
