Usually when that happens there's a way to tie it back to circles, but it's not always easy to find
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.
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?
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
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/n^2^) which converges to π^2^/3
So I can see why it converges, just not where to.
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.
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)
Not really. The left side comes from trying to evaluate the Reimann zeta function at a point outside its domain, and the right side comes from evaluating the analytic continuation of the zeta function at the same point. The deepest truth it reveals is that applying the definition of a function outside its domain can give you nonsense results.
It's one of those ideas like Scrödinger's cat or pulling yourself up by your bootstraps, where something that was intended to illustrate an absurdity is instead taken as representing some amazing and unintuitive truth.
And 2 is.
Tired of waiting 2 seconds for hyperpi to download? Simply do infinity calculations by hand and some basic algebra. Cut out the middle man. Thanks, internet!
omg I missed the ... and was confused as heck
thanks for using Leebra!
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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
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