  We study the behavior of Lagrangian perturbative solutions. For a spherical
void model, the higher order the Lagrangian perturbation we consider, the worse
the approximation becomes in late-time evolution. In particular, if we stop to
improve until an even order is reached, the perturbative solution describes the
contraction of the void. To solve this problem, we consider improving the
perturbative solution using Shanks transformation, which accelerates the
convergence of the sequence. After the transformation, we find that the
accuracy of higher-order perturbation is recovered and the perturbative
solution is refined well. Then we show that this improvement method can apply
for a $\Lambda$CDM model and improved the power spectrum of the density field.
