Make the trace of Tr[exp(-(hat(H))/(k_B)T)] = ∑_(n=0)([infinity]) (:n|exp(-(hat(H))/(k_B)T)|n:)
The final result is Z = exp(-ℏ(ω)/(2)k_B T) ∑_(n=0)([infinity]) exp(-ℏω(n)/(k_B)T)
a) True
b) False