# Rambling Thoughts

## Problems

A rabbit climbs out at its hole, and walks $1$ mile in a straight line. Then, the rabbit repeatedly turns $\pi/3$ radians and walks half of the distance it just walked,
as pictured below.

How far away from the rabbit’s hole is the point at which the rabbit converges?

### 4 responses to “Problems”

1. Américo Tavares June 13, 2011 at 4:25 am

The distance is $\frac{2}{3}\sqrt{3}$ miles. Assume the rabbit starts at the origin of the complex plane and walks on it. The rabbit’s walk can be modeled by the series

$\displaystyle\sum_{n=0}^{\infty }\frac{1}{2^{n}}e^{in\pi /3}$,

which converges to $z=1+\frac{1}{\sqrt{3}}i$ (according to Wolframalpha). So the distance to the origin is $\left\vert z\right\vert =\frac{2}{3}\sqrt{3}$.

2. Américo Tavares June 13, 2011 at 4:32 am

I’ve now noticed that you have already posted a solution.

3. Américo Tavares June 13, 2011 at 7:14 am

After reading your post I recognize that the above series is a geometric series with ratio $\dfrac{1}{2}e^{i\pi/3}$.