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
Don’t think of it as half the time LESS, think of it as half the time LEFT. Thus, no matter how many rooms there are, you never use all of the time available.
The maid starts with 60 seconds left to clean all the rooms. Half of the time left is 30 seconds so, for the first room, she takes 30 seconds. Now there’s 30 seconds left so for the second room, she takes 30/2 = 15 seconds. Repeat and repeat as often as you like, but you never actually take ALL of the time left. She’ll get ever closer to that one minute mark but would never actually reach it.
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.
Take a square of area one. This is
* 1 = 1
Draw a line down the middle to cut it in half. This will give
* 1 = 1/2 + 1/2
Take one of these halve and draw a line down its middle to cut it in half. So one of the halves becomes two pieces of 1/4. So
* 1 = 1/2 + 1/4 + 1/4
Do this again with one of the fourths and you get
* 1 = 1/2 + 1/4 + 1/8 +1/8
Repeat this *ad infinitum* and you’ll get a whole square with successively smaller halves cut into half and so you get
* 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + …
Note, there is no 1/infinity since this halving never ends. You can visualize it [like this](https://francisbach.com/wp-content/uploads/2019/12/triangles-1-1024×447.png).
With infinite series, if the series has a limit, then the series is equal to that value. The limit of 1/2 + 1/4 + 1/8… is 1, and so the infinite series equals 1. If the series has no limit, then it is equal to infinity.
A fun (controversial) series is the sum of all natural numbers. 1 + 2 + 3… It has no limit, and therefore should be equal to infinity, but there are proofs that prove it equals -1/12. Some advanced physics only work if the answer is -1/12, so that HAS to be the answer. But how do you get a negative fraction by adding positive whole numbers? Infinity is fun.
If you have the infinite sum….
“S = 1/2 + 1/4 + 1/8 + … + 1/(2^n)“
Then multiplying it by two you find….
“2*S = 1 + 1/2 + 1/4 + 1/8 + … + 1/(2^n)“
Then subtracting one from the other you get….
“2*S – S = 1 + (1/2 + 1/4 + 1/8 + … + 1/(2^n)) – (1/2 + 1/4 + 1/8 + … + 1/(2^n))“
“S = 1“
Which you can see because all of the terms (except for the 1 at the beginning) get subtracted off again.