# A divergent series from Futurama

I was watching Futurama and in a recent episode, the professor creates a duplication machine.

The machine basically took something and then made 2 copies at 60% the size.

Somehow Bender got caught in the machine and he started duplicating infinitely.

The problem is, Bender needed excess matter to create these duplicates.

The professor then reveals this equation which would explain why it was a threat to the universe as the ammount of matter needed to create the excess bending units would never converge to 0 thus, eventually, needing the entire mass of the universe to keep the series going:

This was already asked on SciFi.SE, but I'm still confused as to what this equation is actually saying. And how it is divergent.

Or if it is even accurate in it's concept.

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If I am reading it correctly, the $2^n$ in the numerator and denominator cancel, leaving behind the famous harmonic series (multiplied by an $M_0$). The wikipedia page discusses the divergence of the series. –  Srivatsan Oct 4 '11 at 4:53
I thought Bender was the robot?? –  Altar Ego Oct 4 '11 at 5:39
For general education: Futurama is usually very accurate and correct about mathematical things. The producer Ken Keeler has a Ph.D. in applied mathematics (he even proved a theorem just for the episode The Prisoner of Benda), and as far as I heard there are a couple of other mathematicians on the staff (this claim, however, I cannot verify anywhere). –  Asaf Karagila Oct 4 '11 at 6:37
There is some relevant discussion at theinfosphere.org/Talk:Benderama, with someone saying that it's never claimed in the episode that all copies are at $60\%$ size, only the first generation of copies, and that the decrease in linear dimensions from halving the mass in the first generation does work out to $\sqrt[3]{1/4}\approx63\%$. (There is further discussion at physicsforums.com/showthread.php?t=509270 and scifi.stackexchange.com/questions/4291/…) –  joriki Oct 4 '11 at 6:50

## 1 Answer

(1) The situation you describe (2 copies at 60% the mass) is a divergent series.

(2) The sum in the picture linked above is also divergent -- it's harmonic.

(3) The sum in the picture does not represent the situation you describe. For that, the sum would look like

$$M = \sum_{n=0}^\infty 2^n M_0 (0.6)^n$$

which is a divergent geometric series.

(4) The head writer on Futurama studied physics at Harvard and CS at Berkeley, has published math papers, and earlier episodes of Futurama have featured much more sophisticated math than this, so it's likely that the equation displayed above accurately represents the situation described in the show. I'll have to try to catch a rerun and see exactly what Farnsworth says.

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If you refer to Ken Keeler, he has a Ph.D. in applied mathematics. –  Asaf Karagila Oct 4 '11 at 6:38
The question says 60% size, not mass. It's not clear from the question whether this refers to linear dimensions or volume. –  joriki Oct 4 '11 at 6:46
Nonetheless, any such way of interpreting the question would lead to a geometric series, not a harmonic series as shown. –  Daniel McLaury Oct 4 '11 at 14:00