= Solution
A $J=1/2$ doublet has moment projections $\pm g_J\mu_B/2$. The requested normalization assumes $g_J=2$, so the projections are $\pm\mu_B$ and the two Zeeman energies are $\mp\mu_BB$. For independent localized ions, the one-ion <partition function> and thermal moment are
$$
Z_1=e^{\beta\mu_BB}+e^{-\beta\mu_BB}=2\cosh(\beta\mu_BB),
$$
$$
\langle\mu_z\rangle=\frac{\mu_B e^{\beta\mu_BB}-\mu_B e^{-\beta\mu_BB}}{Z_1}
=\mu_B\tanh(\beta\mu_BB).
$$
Therefore the <spin-one-half paramagnetic magnetization> is
$$
\boxed{M=n\mu_B\tanh\left(\frac{\mu_BB}{k_BT}\right).}
$$
It saturates at $n\mu_B$ for a field strong compared with $k_BT/\mu_B$. For a different <Landé g-factor>, replace every moment $\mu_B$ by $g_J\mu_B/2$; $J=1/2$ alone does not fix that factor.
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