Comment on Magic

grubberfly@mander.xyz ⁨1⁩ ⁨month⁩ ago

• BugleFingers@lemmy.world ⁨1⁩ ⁨month⁩ ago

  I believe starting with 1/1 which equals 1, you are then adding infinitely (fractions) on top of the 1. So 1, then 1 1/2, ect, so the next full integer to be hit (infinitely down the line) would be 2.

  I don’t do high level math so I hope this explanation is correct or intelligible, this is just how I understand it intuitively

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  • yetAnotherUser@discuss.tchncs.de ⁨1⁩ ⁨month⁩ ago

    But the first few values are:

    1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + 1/28…

    I really don’t see any pattern there showing why it converges to 2 exactly

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    • Bender_on_Fire@lemmy.world ⁨1⁩ ⁨month⁩ ago

      I didn’t see the pattern either and had to look it up. Apparently, you can rewrite 1 + 1/(1+2) + 1/(1+2+3)+… as 2(1 - 1/2 + 1/2 - 1/3 +…+1/n - 1/(n + 1)) = 2(1 - 1/(n + 1))

      From there, the limit of 2 is obvious, but I guess you just have to build up intuition with infinite sums to see the reformulation.

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    • BugleFingers@lemmy.world ⁨1⁩ ⁨month⁩ ago

      So the amount you are adding is getting smaller with each iteration, 1/4 is smaller than 1/2, however you are still adding 1/4 on top of the 1/2, and those two are combined, closer to “1” than either of them independently correct? (1/2 +1/4 =1/3. 1/3>1/2)

      So if the number gets bigger forever than at some point it will eventually hit “1”, since we already started with “1” the next “1” will be “2”

      I hope I’m explaining it well enough, it’s similar to how 3.33(repeating)x3…=10 (though technically for different reasons)

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    • criticon@lemmy.ca ⁨1⁩ ⁨month⁩ ago

      Those add to 1.75, just keep adding (infinitely)

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    • criitz@reddthat.com ⁨1⁩ ⁨month⁩ ago

      You’re always adding 1/3 of the remaining distance to 2

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• lolcatnip@reddthat.com ⁨1⁩ ⁨month⁩ ago

  It’s not saying the proof is obvious, just that the result is plausible on its face.

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• Collatz_problem@hexbear.net ⁨1⁩ ⁨month⁩ ago

  1/(n * (n+1)) = 1/n - 1/(n+1)

  1/(12) + 1/(23) + 1/(3*4) + … = 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + … = 1

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  • AernaLingus@hexbear.net ⁨1⁩ ⁨month⁩ ago

    Written a bit more explicitly (although I kinda handwaved away the final term–the point is that you end up with one unpaired term which goes to zero)

    Image

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