Spins in an external magnetic field

Jolb

This is from Kardar's Statistical Physics of Particles, p.123, question 8.
1. The problem statement, all variables and given/known data
Curie suceptibility: consider N non-interacting quantized spins in a magnetic field [tex]\vec{B}=B\hat{z}[/tex] and at temperature T. The work done by the field is given by BM_z, with a magnetization
[tex]M_z=\mu\sum_{i=1}^{N}m_i[/tex]. For each spin, m_i takes only the 2s+1 values -s, -s+1, ..., s-1, s.

[Parts a-c omitted.]

d) Show that
[tex]C_B -C_M = c \frac{B^2}{T^2}[/tex], where C_B and C_M are heat capacities at constant B and M, respectively.


2. Relevant equations
I'm not totally sure the following equations are absolutely correct, since I've tried using them to no avail. Some of them are the ones I derived in parts a-c and others are ones from Kardar. So I'll separate the ones I found from the ones in Kardar, and for the ones in Kardar, I'll include where Kardar puts them if you want to check if they apply.
Kardar equations:
[tex]G=G=-k_B T\mathrm{ln}Z [/tex](Kardar 4.88)
[tex]M = -\frac{\partial G}{\partial B}[/tex] (Kardar 4.97)
[tex]H = -\frac{\partial \mathrm{ln}Z}{\partial \beta }[/tex] where β=k_BT (Kardar 4.89)
[tex]C_M=\frac{\partial H}{\partial T}[/tex] (given in Kardar just below 4.89)
[tex]C_B=-B\frac{\partial M}{\partial T}[/tex] (given in Kardar just below 4.98)

My equations:
[tex]Z=\left (\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}} \right )^N[/tex]
[tex]G = -Nk_B T\mathrm{ln}\frac{1-e^{\frac{-(2s+1)B\mu}{k_B T}}}{1-e^{\frac{-B\mu}{k_B T}}}[/tex]


3. The attempt at a solution
I've written down C_B and C_M but I have exponentials hanging around which don't cancel and therefore don't leave me with B^2/T^2 proportionality. I suspect there's something wrong with the results of my a-c, which are all in "my equations."


Thanks.