Infinitely Dividing the Pie
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A tasty meal-time lesson in summing infinite geometric series.
Contributed by: Kenneth E. Caviness (June 2019)
Open content licensed under CC BY-NC-SA
A simple meal-time illustration so that even a child can understand the fundamental idea that adding up an infinite number of terms can still result in a finite value, as long as the terms are decreasing fast enough.
Suppose you have five people at the table, all looking forward to sharing a delicious, home-baked pie. Unfortunately, no one at the table is able to divide the pie into five equal pieces, and cutting it in six pieces seems easier. All present take a piece, but that leaves one piece unclaimed. Common sense now suggests several options: give the remaining piece to the hungriest teenager present, or divide it between those who want a little extra. But all five people want to share this last piece equally, and they still cannot figure out how to divide it fairly, so they end up cutting it in six equal pieces, each taking one, leaving one left over—and yes, the process can be repeated forever! (Or at least until the remaining piece is too small to divide even under a microscope.) Clearly, all five people have had equal shares of pie, so this process of dividing in sixths, repeatedly, has really divided the pie in fifths. In other words, a sixth plus a sixth of a sixth, plus a sixth of a sixth of a sixth, plus … equals a fifth.
Expressed as a formula,
This is a special case of the sum of a geometric series, with first term and the ratio of adjacent terms . The sum is then
The same example illustrates that the sum of the geometric series with and is . (Eating of the pie and then , followed by …, means that in the limit, the whole pie is gone.)
In this Demonstration, you can select , the number to repeatedly divide by (one more than the number of people at the table), how many iterations of the process to show, and whether or not to sort the pieces, bringing all of each person's pie together to make the sum more meaningful visually. Two- and three-dimensional views can be selected.
Snapshot 1: the first iteration of dividing the pie into quarters:
Snapshot 2: the first two iterations of dividing the pie into quarters:
Snapshot 3: the first four iterations of dividing the pie into quarters:
Snapshot 4: the first four iterations of dividing the pie into quarters, sorted to show that each person's total piece of the pie is approaching one third
Snapshot 5: repeatedly dividing in half demonstrates the sum of geometric series