Joshua, That would be nice, because I am pretty averse to generating functions too. So can you show all the steps so it is clear what you mean? What do you multiply by 10000 and what do you subtract from what? And how do you end up with the numerator and denominator 100010000 / 999700029999? Does your method work for any integer sequence? How much do I need to know about the definition of the integer sequence or how to generate it? - Robert On Sun, Jan 29, 2012 at 03:58, Joshua Zucker <joshua.zucker@gmail.com>wrote:
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
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com