My pre-Calculus students were having trouble visualizing what it meant for a geometric series to converge. They understood the formula aspect and how to calculate the value of a convergent sum, and saw what it meant for numbers adding successively, but I knew there had to be a way to show them.
I had visited Russia with my wife during the winter holidays and saw some amazing arrangements of stacking dolls. I thought at the time about the concept of common ratio, but didn't think about its application to series. Later on, when Dan Meyer posted about a video about stacking dolls (<url>http://blog.mrmeyer.com/?p=10211</url>) I realized this was a perfect way to present and explore convergence.
In order to spice it up just a little bit more (and it was worth the entertainment value), I made the 'object' being stacked in Geogebra an image of my own head. The students saw pretty quickly when the series converged based on the common ratio value, and it seemed like a fun way for them to confirm it in class.
