Ok so the problem: 2 trains are 100 miles apart traveling towards each other on the same track. Each train tavels at 10 miles per hour. A fly leaves the first train heading towards the second train the instant they are 100 miles apart. The fly travels at 30 miles per hour (relative to the ground not relative to the train he left). When the fly reaches the second train, it turns and heads back to the first train at 30 miles per hour (assume that the change in direction takes zero time). when the fly reaches the first train, he turns again. this process continues with the fly zipping back and forth between the trains as they come ever closer, until the two trains colide. The question: How far does the fly travel until he is crushed? (total distance traveled not displacement from original position)
It's also a famous anecdote When this problem was posed to John von Neumann, he immediately replied, "xxx miles." "It is very strange," said the poser, "but nearly everyone tries to sum the infinite series." "You mean there's another way?" says von Neumann!
Well the trains will meet in five hours so if the fly flies continuously for five hours, at 30mph, he travels 150 miles. mathematically, [tex]d=\int \limits_0^5 30~dx=30\int \limits_0^5 ~dx = 30[x]\limits_0^5 = 30[5-0] = 150 miles[/tex]
When I first heard this problem many years ago that how I did it. I however also realized the infinite geometric approach but it is too computational. I do not find this to be challenging problem and I am sure most will agree.
The way I heard it after hearing the problem, Von Neuman had the answer immediately. "Wonderful! You must have seen the simple way! Most people take forever trying to sum the infinite series!!" "But I used infinite series."