Ive just watched the Netlfix documentary called A Trip To Infinity, they mentioned the very famous infinity hotel paradox but with something extra i have not heard yet.. they said the maid is done checking all the rooms in the infinite hotel in ONE MINUTE, by taking half the time less in the next room. 30 seconds in room 1, 15 seconds in room 2, 7,5 seconds in room 3. So 1/2 + 1/4 + 1/8…….. + 1/infinity equals ONE. Could someone please explain the math behind this, thank you
In: 4
At risk of stating the obvious, you cannot actually check infinite hotel rooms in a minute. The idea here is that if *theoretically* you could do each room in half the time as the previous rooms, the total sum of all the times would be one minute.
Let me illustrate with an easier example. Let’s say I move infinitely many steps, but the distance of each step is precisely a tenth of the last. Suppose my first step was 1 meter, then the next step would be 0.1 meters, then 0.01 meters, 0.001 meters, etc.
If I want to know how far I’ve gone after N steps, I just have to add together the distances I’ve moved. This would be
* 1.1 meters after the second step
* 1.11 meters after the third step
* 1.111 meters after the fourth step
* 1.111111111 meters after the tenth step
* and so on…
It’s easy to see that no matter how many steps I take, I’m never going to make it to 2 meters following this pattern. In fact, after *infinitely many* steps, I’ll have moved precisely 10/9 meters.
This is the mathematical idea of a *convergent sum*. Basically you can add together infinitely many numbers and get a finite number, as long as those numbers are approaching zero quickly enough. Figuring out what “quickly enough” means is a good chunk of what calculus, and later analysis, is about.
Halving each step, or dividing each step by 10 certainly forces each step to approach zero quickly enough, to the point of being pretty overkill.
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