Expansion formulas in Riemannian geometry and normal coordinates

Question

In normal normal coordinates systems on a Riemannian manifold $(M,g)$ you get local expansion formulas. For example the metric is given by $$ g_{\mu\nu}(x) = \eta_{\mu\nu} - \tfrac{1}{3} R_{\mu\alpha_{1}\alpha_{2}\nu} x^{\alpha_{1}} x^{\alpha_{2}} + \tfrac{1}{6} R_{\mu ( \alpha_{1}\alpha_{2} | \nu ; | \alpha_{3} ) } x^{\alpha_{1}} x^{\alpha_{2}} x^{\alpha_{3}} + \tfrac{1}{20} \bigl[ R_{\mu ( \alpha_{1}\alpha_{2} | \nu ; | \alpha_{3} \alpha_{4} ) } + \tfrac{8}{9} R_{\mu ( \alpha_{1}\alpha_{2} | \delta} R^{\delta}{}_{ | \alpha_{3}\alpha_{4} ) \nu} \bigr] x^{\alpha_{1}} x^{\alpha_{2}} x^{\alpha_{3}} x^{\alpha_{4}} + \mathcal{O}(x^{5}) .$$ My problem is to understand what the remainder term $O(...)$ means. Does this expansion formula imply that there is a way to find a constant $C$ independent of the geometry such that $$ |g_{\mu\nu}(x) - \eta_{\mu\nu}| \leq C \sup_{x \in M} |R| |x|^2, $$ where $R$ is some norm of the curvature tensor $R$? Is there a reference for these kinds of estimates?