The Cantor's Disappearing Table is a famous and super fun series problem regarding a table and whether it'll disappear by cutting only half of the table. Sounds incredibly absurd but it is super interesting to solve! Lets explore this problem in depth: We will start with a table, with the length L. Lets remove 1/4 of the table from the center, leaving two pieces in the sides. The remaining pieces each definitely have a length less than 1/2L. Then, lets remove 1/8 of the table by removing 1/16th from each of the remaining sides. Each of the remaining pieces have a length less than 1/4L. Next, we remove 1/16th of the desk by removing 1/64th from each previous remaining piece. Each of the remaining pieces have a length less than 1/8L. As we continuously remove this portions of the table, will the table eventually disappear? There are two ways to look at this problem. One way is focusing on the amounts removed from the table, and the other way to look at this problem is focusing on the amounts remaining. Amounts RemovedSo if we continuously remove the portions of the table in this pattern, the series works out to be: 1/4L + 1/8L +1/16L + 1/32L + .... = (n=2 to ∞) ∑ (1/2^n)L = 1/2L. This means, that we will actually remove half of the table by following this process. I obtained 1/2L by: S = 1/4L + 1/8L +1/16L + 1/32L + .... S = (1/2)^2 + (1/2)^3 + (1/2)^4 + (1/2)^5... = (1/2)^2 + 1/2( (1/2)^2 + (1/2)^3 + (1/2)^4) = S = (1/2)^2 + 1/2(S) 1/2S = 1/4 S = 1/2 So this leads to the conclusion that only half of the table will ultimately disappear but the entire table won't. Seems really absurd, but it is mathematically proven to be correct in this way. Amounts RemaningSo if we focus on the amounts remaining, we will end up with a different conclusion: that the table does eventually disappear. Lets make a series for the remaining pieces: S = 1/2L + 1/4L +1/8L + 1/16L + .... In order to find out how much is remaining: ; lim(as n approaches ∞) (1/2)^n = (1/2)^∞ = 0. So this means that the remaining lengths will approach 0L. This means that the pieces will ultimately have a length of 0. This proves that the table eventually disappears. We can conclude that there are two different sides to this problem. It just depends on which way you choose to look at it, whether it is the remaining pieces or the removed pieces. This problem was so fun to solve and it was so interesting to prove the two different perspectives!! Good luck as always!
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