[When \(x_0\ =\ 0\), this is known as a Maclaurin series.] Note that since the system is in equilibrium at \(x=x_0\), we have \(U'\left(x_0\right)\ =\ 0\). If we consider "small" oscillations, we can ignore terms in \(\left(x-x_0\right)^3\) or higher powers, as long as the coefficient of the \(\left(x-x_0\right)^2\) term is non-zero. For a stable equilibrium, we need \(U''\left(x_0\right)>0\). If so, our approximation simplifies to a parabolic potential: