  We show that by means of connected-graph expansions one can effectively
generate exact high-order series expansions which are informative of low-lying
excited states for quantum many-body systems defined on a lattice. In
particular, the Fourier series coefficients of elementary excitation spectra
are directly obtained. The numerical calculations involved are straightforward
extensions of those which have already been used to calculate series expansions
for ground-state correlations and $T=0$ susceptibilities in a wide variety of
models. As a test, we have reproduced the known elementary excitation spectrum
of the transverse-field Ising chain in its disordered phase.
