In Riemann normal coordinates (which, by definition, are the coordinates of a local inertial frame) the equation of a geodesic through the origin $p$ is
$x^\mu(s) = a^\mu s$ where $a^\mu = \frac{dx^\mu} {ds}|_p$ and $s$ is a real curve parameter.
I have derived from the geodesic equation that
$\frac{d^3 x^\mu}{ds^3}=-\partial_\rho\Gamma^\mu_{\nu\lambda}|_p a^\nu a^\lambda a^\rho$
and that
$\partial_{(\rho}\Gamma^\mu_{\nu\lambda)}|_p=0$
How do I prove from the last equation and using Taylor expansion of the coefficients of the metric at the origin that:
$g_{\mu \nu}= g_{\mu \nu}|_p + \frac{1}{3}R_{\lambda \mu \nu \rho}|_p x^\lambda x^\rho +O(x^3)$
?