On Tuesday 19 January 2010 17:17:42 Dan Asimov wrote:
Let a, b, m > 0, with m < min{1/a, 1/b}. ... Repeat infinitely many times, obtaining R(2), L(2), etc. What is the total length of all the segments so drawn?
"I summed the series." -- John von Neumann. [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] [SPOILER SPACE] Seriously: total vertical displacement is 1. Total length equals length per unit vertical displacement equals sqrt(1+1/m^2). No? (Note that you get the same result wherever you start on the x-axis. Similarly, it doesn't matter where the fly starts; it always travels the same distance before it gets squashed.) -- g -- g