  We analyze the large-order behavior of the perturbative weak-field expansion
of the effective Lagrangian density of a massive scalar in de Sitter and anti
de Sitter space, and show that this perturbative information is not sufficient
to describe the non-perturbative behavior of these theories, in contrast to the
analogous situation for the Euler-Heisenberg effective Lagrangian density for
charged scalars in constant electric and magnetic background fields. For
example, in even dimensional de Sitter space there is particle production, but
the effective Lagrangian density is nevertheless real, even though its
weak-field expansion is a divergent non-alternating series whose formal
imaginary part corresponds to the correct particle production rate. This
apparent puzzle is resolved by considering the full non-perturbative structure
of the relevant Feynman propagators, and cannot be resolved solely from the
perturbative expansion.
