Solution
= Solution
In the continuum limit the number outside the ground state is
$$
N_{\rm ex}=BV\int_0^∞{E^{d/α-1}\over e^{β(E-μ)}-1}\,dE.
$$
At the largest allowed chemical potential, $μ=0$, the integrand behaves near zero as $E^{d/α-2}$. The integral is finite exactly when $d/α>1$. Excess particles must then occupy the ground state, so a <Bose-Einstein condensation>[Bose-Einstein condensate] exists precisely for $α<d$.