  We calculate the energy of the state closest to threshold for two and three
identical, spinless particles confined to a cubic spatial volume with periodic
boundary conditions and with zero total momentum in the finite-volume frame.
The calculation is performed in relativistic quantum field theory with
particles coupled via a $\lambda \phi^4$ interaction, and we work through order
$\lambda^3$. The energy shifts begin at ${\cal O}(1/L^3)$, and we keep
subleading terms proportional to $1/L^4$, $1/L^5$ and $1/L^6$. These terms
allow a non-trivial check of the results obtained from quantization conditions
that hold for arbitrary interactions, namely that of L\"uscher for two
particles and our recently developed formalism for three particles. We also
compare to previously obtained results based on non-relativistic quantum
mechanics.
