If we add to this the corresponding expansion for the derivative of the velocity $$v^{(1)}_{i+1} = v^{(1)}_i+hv^{(2)}_i+O(h^2),$$ and retain only terms up to the second derivative of the velocity since our error goes as $O(h^3)$, we have $$hv^{(2)}_i\approx v^{(1)}_{i+1}-v^{(1)}_i.$$ We can then rewrite the Taylor expansion for the velocity as $$v_{i+1} = v_i+\frac{h}{2}\left( v^{(1)}_{i+1}+v^{(1)}_{i}\right)+O(h^3).$$