= Solution
<Hydrogen>'s $n=2$ <excitation energy> is $3R/4=10.2$ eV. With electronic degeneracy ratio $g_2/g_1=4$, its relative population is $4e^{-10.2/1.60}\simeq6.8\times10^{-3}$. Thus the first excited manifold contributes below one percent of the ground population at the inferred <temperature>, explaining the leading ground-state approximation.
There is a qualification to extending this <argument> to infinitely many Rydberg states: the isolated-atom partition sum with degeneracy $2n^2$ formally diverges. <Plasma screening> and level dissolution impose a physical cutoff. The small low-lying populations justify the simple approximation in its intended regime, not an unrestricted infinite partition sum.
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