Answer
With $N$ atoms per unit volume, we find the magnetization to be
$$
M=\frac{N \mu P(\mu)-N \mu P(-\mu)}{P(\mu)+P(-\mu)}=\frac{N \mu\left(e^{\mu B / k T}-e^{-\mu B / k T}\right)}{e^{\mu B / k T}+e^{-\mu B / k T}}\\=N \mu \tanh \left(\frac{\mu B}{k T}\right) .
$$
Work Step by Step
With $N$ atoms per unit volume, we find the magnetization to be
$$
M=\frac{N \mu P(\mu)-N \mu P(-\mu)}{P(\mu)+P(-\mu)}=\frac{N \mu\left(e^{\mu B / k T}-e^{-\mu B / k T}\right)}{e^{\mu B / k T}+e^{-\mu B / k T}}\\=N \mu \tanh \left(\frac{\mu B}{k T}\right) .
$$
