Chapter 5, Problem 29RE

To determine

To show: The power series solution of the differential equation $\frac{d^2\theta }{d{t}^2}+\frac{g}{l}\sin \theta =0$, $\theta \left(0\right)=\frac{\pi }{6}\text{ and }{\theta }^{\prime }\left(0\right)=0$ is $\theta \left(t\right)=\frac{\pi }{6}-\frac{g}{4l}{t}^2+\frac{\sqrt{3}{g}^2}{96l^2}{t}^4+\cdots$.