If you want to be convinced that you can do it later, I think I've managed to go through it correctly.
Let x =(beta * g * B * mu_b).
The partition function can be written as (e^Jx - e^(J+1)x)/(1-e^x) = sinh(x(J+1/2))/sinh(x/2)
-dF/dB = k_B*T(coth((x/B)(J+1/2)) - coth ((x/B)(J-1/2)))
Making the small B expansion we have -dF/dB = k_B*T[1/B - 1/B + (x^2/3B)*((J+1/2)^2 - (1/2)^2)]
And then just differentiate wrt B and multiply by n*mu_0.
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