On Sat, Jan 28, 2012 at 10:23 PM, Robert Munafo <mrob27@gmail.com> wrote:
What rational fraction is equal to .000100040009001600250036...?
You can also sum the series n^2 / 10000^n in other ways (for example: repeatedly multiply by 10000 and subtract), if you want to make it accessible to people who don't know about generating functions (like the 10th graders I work with). What I'm trying to understand is the length of the period of these things. I mean, it is a list of squares, but eventually carries start messing things up, which intuitively seems like it would make it not a repeating decimal, but I'm staring at the fractional form of it! How can you account for the carries and explain how many digits it takes for these things to repeat by analyzing the sequence of squares rather than the denominators? --Joshua Zucker