I understand that the equation for kinematic displacement is:

$x = v_{0x}t+\frac{1}{2}a_xt^2$

Perhaps my understanding is naive, but it seems like this leaves out higher order rates of change. Why wouldn't the equation be like:

$x = v_{0x}t+\frac{1}{2}a_xt^2+\frac{1}{6}j_xt^3+\frac{1}{24}s_xt^4+\frac{1}{120}c_xt^5+. . . $

where $j_x$ represents jerk, $s_x$ represents snap, $c_x$ represents crackle, and so on for $n$ number of higher-order terms, perhaps as an infinite series?