Start with an angle of $60^{\circ}$ and bisect it, then bisect the lower $30^{\circ}$ angle, then the upper $15^{\circ}$ angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original $60^{\circ}$ angle into two angles. Find the measure (degrees) of the smaller angle.
Solution: The fraction of the original angle is $\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-+\cdots$. This is just a geometric series with first term $1 / 2$ and ratio $-1 / 2$, so the sum is $1 / 3$. Therefore the smaller angle is $\mathbf{2 0}^{\circ}$.