  We calculate the average differential entropy of a $q$-component Gaussian
mixture in $\mathbb R^n$. For simplicity, all components have covariance matrix
$\sigma^2 {\mathbf 1}$, while the means $\{\mathbf{W}_i\}_{i=1}^{q}$ are i.i.d.
Gaussian vectors with zero mean and covariance $s^2 {\mathbf 1}$. We obtain a
series expansion in $\mu=s^2/\sigma^2$ for the average differential entropy up
to order $\mathcal{O}(\mu^2)$, and we provide a recipe to calculate higher
order terms. Our result provides an analytic approximation with a quantifiable
order of magnitude for the error, which is not achieved in previous literature.
