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
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.
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