• @[email protected]English
    12•2 months 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.

    • @[email protected]English
      6•2 months 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.

    • BuglefingersEnglish
      2•2 months 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)