Clear[linPart, quadPart, z, zv, cv, s1, s2]

linPart[ u_] = Sum[ E^(-u (n + 1/2)),{n,0,Infinity}] // FullSimplify ;

z[ u_, x_] = linPart[ u] + x u D[ linPart[u], {u, 2}] // FullSimplify

cv[u_, x_] = u^2 D[Log[z[u, x]], {u, 2}] // FullSimplify 

s1[u_, x_] = Series[cv[u, x], {x, 0, 1}] 
s2[u_, x_] = Series[ s1[u, x], {u, 0, 4}] 

Manipulate[
Plot[cv[1/t, x], {t, 0, 1.5}, PlotRange -> Full, AxesLabel->{(Subscript[k, B])T/(ℏ ω), (Subscript[C, V])/(N (Subscript[k, B]))}]
, {{x, 0.1},0.001,0.1}]