= Solution
Set $q=e^{-\beta\hbar\omega}$. The oscillator populations are $(1-q)q^n$, and differentiating the <geometric series> gives $\bar n=q/(1-q)$. The <canonical oscillator coordinate variance> follows from $2\bar n+1=(1+q)/(1-q)=\coth(\beta\hbar\omega/2)$, and
$$
\boxed{\langle x^2\rangle_T=\frac{\hbar}{2m\omega}\coth\left(\frac{\hbar\omega}{2k_BT}\right)}.
$$
It approaches $\hbar/(2m\omega)$ at zero <temperature> and $k_BT/(m\omega^2)$ at high <temperature>, agreeing respectively with ground-state fluctuations and classical equipartition.
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