  We consider Eisenstein series appearing as coefficients of curvature
corrections in the low-energy expansion of type II string theory four-graviton
scattering amplitudes. We define these Eisenstein series over all groups in the
E_n series of string duality groups, and in particular for the
infinite-dimensional Kac-Moody groups E9, E10 and E11. We show that,
remarkably, the so-called constant term of Kac-Moody-Eisenstein series contains
only a finite number of terms for particular choices of a parameter appearing
in the definition of the series. This resonates with the idea that the constant
term of the Eisenstein series encodes perturbative string corrections in
BPS-protected sectors allowing only a finite number of corrections. We underpin
our findings with an extensive discussion of physical degeneration limits in
D<3 space-time dimensions.
