  We consider the generalised Mathieu series \[\sum_{n=1}^\infty
\frac{n^\gamma}{(n^\lambda+a^\lambda)^\mu}\qquad (\mu>0)\] when the parameters
$\lambda$ ($>0$) and $\gamma$ are even integers for large complex $a$ in the
sector $|\arg\,a|<\pi/\lambda$. The asymptotics in this case consist of a {\it
finite} algebraic expansion together with an infinite sequence of increasingly
subdominant exponentially small expansions. When $\mu$ is also a positive
integer it is possible to give closed-form evaluations of this series.
Numerical results are given to illustrate the accuracy of the expansion
obtained.
