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There are countless of anecdotes about John von Neumann and his dazzling intellect, one of which is the following:

"Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of the trains flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown?" In a strict mathematical sense the fly actually hits each train an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. This is they way that most trained mathematicians will solve the problem. Conversely a mathematical novice will most likely solve the problem the easy way - since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide, therefore the fly was flying for two hours, at a rate of 75 miles per hour, and so the fly must have flown 150 miles. Easy. When this problem was posed to John von Neumann, he immediately replied, "150 miles." "Ah, I see you've heard this one before, Professor von Neumann. Nearly everyone tries to sum the infinite series." "What do you mean?" asked von Neumann. "That's how I did it!"

I have no experience with infinite series (yet), and my question is:

What infinite series would you obtain from this question?


marked as duplicate by Chris Eagle, Hans Lundmark, Davide Giraudo, Nate Eldredge, Sasha Feb 24 '13 at 15:22

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    $\begingroup$ See the excellent answer by Marvis here. $\endgroup$ – Ragib Zaman Feb 24 '13 at 12:40
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    $\begingroup$ I've heard it with Norbert Wiener instead of John von Neumann. $\endgroup$ – mrf Feb 24 '13 at 13:53
  • $\begingroup$ Since this ask for the series, I think it's a duplicate. $\endgroup$ – leo Feb 24 '13 at 15:20