In a recent hilarious episode of the TV cartoon Futurama, the robot Bender acquired an attachment that allowed him to make 2 duplicates of himself, each 60% of his size, consuming an equal mass of nearby matter in the process for their raw materials. When the professor asked him to do some work, he was lazy, so activated this machine to make 2 smaller replicas of himself to do it. But then these replicas each contained a smaller copy of the duplicating machine as well, and they were just as lazy as the original, so they could make 2 copies each of themselves. You can see where this is going: since all the Benders were equally lazy, they kept duplicating, potentially on to infinity. The alarmed professor flashed an equation on the screen, and everyone gasped. "The equation is divergent", he explained, which meant that soon replicating Benders would consume all the matter on Earth! What was he talking about?
Let's review the concept of convergent and divergent series. Suppose you are adding together an infinite series of smaller and smaller numbers: say, 1/2 + 1/3 + 1/4 ... . There are two things that could happen. Either the series is divergent, in which case the sum approaches infinity, or it's convergent, in which case the total is some finite number. At first, it might seem counter-intuitive that the sum of an infinite series could ever do anything but diverge to infinity. But here's a simple counterexample: Look at the decimal number .99999..., with an infinite number of nines after the decimal point. I think we would all agree at a minimum that this is a finite value, less than or equal to 1. (It's actually precisely equal to 1, but we'll leave that nuance for another podcast.) In any case, if you look at each digit of the number separately, you can see it is just an infinite series: the first 9 after the decimal represents 9/10, the 2nd represents 9/100, and so on. So .99999... is the same as the infinite series 9/10 + 9/100 + 9/1000..., and we know that despite the infinite number of terms, the sum never gets past 1. Similarly, if the total mass of the infinite Benders converges to a small finite number, the Earth is not doomed after all.
So, let's take another look at the problem of the replicating Benders. Rather than kilograms, let's simplify our calculations and measure mass in Bender-masses, or Bs, where the mass of the original Bender is 1B. What is the mass of the two mini-Benders he creates? First we have to define what '60% of the size' means: I think it would be logical to assume we are reducing the measure to .6 times the original in each of the 3 dimensions: length, width, height. Assuming the mass is proportional to the volume, this means that the mass of each mini-Bender is .6*.6*.6 Bs, or .216 times the mass of the original Bender. So, the total mass of the two half-Benders is 2*.216 = .432 Bs. Similarly, the Nth generation of Benders should have mass of (.432)^N Bs, since its mass is .432 times that of the previous generation. And the sum we're dealing with is (1 + .432 + .432^2 + ...). Good news-- this series converges! The easiest way to see that is to notice that .432 is less than 1/2, so the Nth term is always less than (1/2^N), a well-known convergent series that adds up to 2. (A quick way to prove this is to look at the base-2 number .1111...) So, no matter how many replicas there are, the total mass will be less than 2 Benders, and our planet is safe.
Unfortunately, the plot of the episode is dependent on the fact that the Bender-masses will add up to a divergent series guaranteed to consume the earth. On the cartoon, the sum the Professor flashes is a series where each term equalled a Bender mass times 2^n * (1 / ((2^n) * (n+1))). You can see here that the 2^n terms cancel out in the top and bottom, making this effectively the sum of 1/(n+1), or 1/2 + 1/3 + 1/4..., a well-known divergent series. If accurate, this would indeed show that all the mass in the universe would ultimately be consumed by the replicating Benders. But where does this equation come from? If the smaller Benders were 60% of the mass instead of 60% of the size in each dimension, then each generation would be 1.2x the mass of the previous, a constantly growing series which obviously diverges, but doesn't match the Professor's series either. I have the feeling the writers are just messing with us, and came up with an arbitrary divergent series to advance the plot. Or maybe the professor messed up the equation; after all, he is often portrayed as a bit senile. On the other hand, I could be the senile one-- tell me if you manage to find something I missed, and think of a good justification for the series in this episode.
And this has been your math mutation for today.