  In quantum mechanics the eigenstates of the Hamiltonian form a complete
basis. However, physicists conventionally express completeness as a formal sum
over the eigenstates, and this sum is typically a divergent series if the
Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can
be reconstructed formally as a sum over its eigenvalues and eigenstates, this
series is typically even more divergent. For the simple cases of the
square-well and the harmonic-oscillator potentials this paper explains how to
use the elementary procedure of Euler summation to sum these divergent series
and thereby to make sense of the formal statement of the completeness of the
formal sum that represents the reconstruction of the Hamiltonian.
