# 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 $\dfrac{1}{2}e^{i\pi/3}$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 7, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9453049898147583, "perplexity": 776.9409552541438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104172.67/warc/CC-MAIN-20170817225858-20170818005858-00698.warc.gz"}