• MarcomachtKuchen@feddit.orgEnglish
    68·
    9 days ago

    I’m allways astonished by how many function seemingly have nothing to do with circles and yet somehow a pi managed to snuck itself in

      • notabot@lemm.eeEnglish
        17·
        9 days ago

        You could say you just go round and round hunting for it, but no matter how hard you try you just can’t corner it.

        Well, you could.

      • xantoxis@lemmy.worldEnglish
        7·
        8 days ago

        there’s a way to tie it back to circles

        Not necessarily circles, but conic sections. When you take a series of a fixed exponent over a variable x, and graph it, that graph is a parabola.

        A parabola is a slice through a cone. Tada, pi appears.

  • Grubberfly 🔮@mander.xyzEnglish
    26·
    9 days ago

    tbf, the 2nd sum is exactly the first one just multiplied by 1/2. though i get that the progression is natural, even, and odd.

    the last one is definitely odd puzzling, but i cannot intuitively get the first one. how does summing the inverse of triangular number equal 2?

    • Buglefingers@lemmy.worldEnglish
      7·
      9 days 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

      • yetAnotherUser@discuss.tchncs.deEnglish
        7·
        9 days 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

        Edit:

        After thinking some more, you could write the sum as:

        (Sum from n=1 to infinity of): 2/(n * (n + 1))

        That sum is smaller than the sum of:

        2 * (1/n2) which converges to π2/3

        So I can see why it converges, just not where to.

        • Bender_on_Fire@lemmy.worldEnglish
          5·
          9 days 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.

        • Buglefingers@lemmy.worldEnglish
          1·
          9 days 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)

  • jeffhykin@lemm.eeEnglish
    6·
    9 days ago

    What’s the big deal? Just solve for …

    … = (π^2 / 6) - (1 + 1/4 + 1/9)

    Ez

    /s