How much horizontal distance can be covered by a stack of blocks? Is it possible to create a stack of blocks that can extend so far sideways that the top block is no longer directly supported by the bottom block? If so, how much farther can the stack go?
This question can be considered in terms of convergence and divergence of a series. If there is a limit to how far the blocks can extend, we could say that this series converges. If not, it diverges. Let's see if we can find a series that describes this situation.
In order to get the greatest possible distances, we will start by thinking about the top block. To make life simpler, let's say that each block is 2 units long. To prevent the top block from tipping over, at most it can only have half of its length sticking out over the edge (that's 1 unit).
The next block down can extend outward until the center of gravity of the top two blocks reaches the edge of the block below them. This adds another half unit (so that's 1 + ½ so far). The third block will extend the stack out an additional 1/3, and the pattern becomes clear.
Step One: Write a Series
Write a series using summation (sigma) notation for the horizontal distance that can be covered by an infinite stack of blocks. (Your answer should look like a familiar series).
Step Two: Converge or Diverge?
Does this series converge or diverge?
The how far could an infinite stack of blocks extend?
Step Three: Horizontal Distance
This horizontal distance covered by the stack can be approximated by ln (N), where N is the number of units in the stack. How many blocks would it take for the stack to extend 2 units, so that the top block is no longer directly supported by the bottom block?