In this paper, we study the approximation orders of a real number [Formula: see text] by the partial sums of its [Formula: see text]-expansions as [Formula: see text] varies in the parameter space [Formula: see text]. More precisely, letting [Formula: see text] be the partial sum of the first [Formula: see text] items of the [Formula: see text]-expansion of [Formula: see text], we prove that for any real number [Formula: see text], the approximation order of [Formula: see text] by [Formula: see text] is [Formula: see text] for Lebesgue almost all [Formula: see text]. Moreover, we obtain the size of the set of [Formula: see text] for which [Formula: see text] can be approximated with a more general order [Formula: see text], where [Formula: see text] is a positive function. We also determine the Hausdorff dimension of the set [Formula: see text] where [Formula: see text] is the number of the longest consecutive zeros just after the [Formula: see text]th digit in the [Formula: see text]-expansion of [Formula: see text].