Remember the post where I talked about the problem with two trains and a bee? No? Alright, I’ll link it here. In that post, I said I would do the “infinite sum” method. In this post, I’ll do just that!
Never mind, I won’t. But let’s use a simpler approach, namely 50kph for each train, 100kph for the bee, and a 300km track.
First trip: the bee starts from end A towards end B. The bee flies twice as fast as a train can move, which is 2:1 for 300km, which is 200km to 100km. That means that after the first trip, the bee will be at 200km from its original, which brings the total distance to 200km so far. Both trains have travelled 100km from their origin, which means there is 100km between the two trains.
Second trip: from train B to train A. In 100km, that will be 2:1 again, which is 66.67km to 33.33km. The bee travels 66.67km, a total now of 266.67km.
Third trip: again, from A to B. With 33.33km left, can you see a pattern? Every new trip, the distance remaining is divided by three, and the distance that the bee travels is 2/3 of that. For any trip, or any sum of trips, the bee will have always travelled 2/3 of the total distance. If a certain trip is 270km, the bee will have travelled 2/3 of that, or 180km. If a group of trips sum to 150km, the bee will have travelled 100km.
Therefore, through repetition and logic, we have deduced that for two trains, both at 50kph, and a bee at 100kph, it will fly for 200 * 3/2 = 300 km.
That brings us to an interesting infinite sum: 1 + 1/3 + 1/9 + 1/27 + 1/81 + … will converge to 1.5, which is very interesting. Why is it 1.5?
Let’s think about base 3. The values are 0, 1, 2, then after 2 is 10. If you add 1, 3 and 9, you get 13. Multiplying by 2 gives 26, one less than the next power of 3, 27. That’s because for any x, 3^x – 1 is 222…2222 in base 3, and adding one will carry everything over. That means, for a sigma sum from x = 0 to n, 3^x is (3^(n+1))/2 – 1, or one less than half of the power of 3 after the final power of three in the sigma sum. Approximately, that’s a half.
When it reaches x = infinity, 3^x is technically double the sigma sum from n = 0 to x-1 of 3^x. Therefore, as x slowly becomes infinity, the sigma sum will become 1.5.
Alright, that’s cool!