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
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).
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