Not -1/12

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blackpenredpen | 曹老師

[tomasouzaheuert]

In a nutshell, the sum of all the positive integers is the friends we made along the way

[thisisreesespieces]

Because infinity is involved, the number can represent anything. It’s like indeterminate form 0/0 and just dividing by 0 in general. Like how there’s one false proof that gets 1=2 by dividing by 0.

[francoiso]

The golden rules to be adhered to when dealing with divergent series are:
1) Do not use brackets
2) Do not remove any zero
3) Do not shuffle around more than a finite number of terms

[cosmos]

As @ViperDaniel points out both -1/12 and -1/8 are related to the sequence 1+2+3+4+... If you graph in the equation (x+1)x/2 (which is the continuous version of the sequence above) you get a parabola. The signed area under the x axis is -1/12, and the minimum point has a height of -1/8. These two numbers are NOT arbitrary, they pop right out from plotting the equation. Saying the word "equals" in these situations is troublesome without giving a very specific context.

[BigDBrian]

I like this as a counterexample to the numberphile method of extracting -1/12

[thesentientneuron]

Thanks blackpenredpen! You're the first person to get me legitimately interested in maths. All my math teachers in school were too slow and boring. But you actually make it interesting.

[nephty]

this man deserves an award for being the master of the pen switch

[non-inertialobserver]

The vertex of the parabola y=x(x+1)/2 is at y=-1/8 LOL
And the area under the x axis (-1 to 0) is -1/12

[dviscusi]

This is kind of interesting in the general case. You always get -1/8.
In general, 2n+1 consecutive integers will sum to a value equal to a multiple of (2n+1)^2, so long as the middle value is a multiple of 2n+1.
Ex.
n = 1: (3k-1)+3k+(3k+1) = 9k
n = 2: = 25k
or in general,
[(2n+1)k - (n)] + [(2n+1)k - (n-1)] + ... + [(2n+1)k - 0] + ... + [(2n+1)k - (n-1)] + [(2n+1)k - (n)] = (2n+1)(2n+1)k = (2n+1)^2 k

This allows for grouping of every (2n+1) terms after the first n terms have been summed, resulting in the nth triangular number.
Ex.
n = 1: (1) + (9)(1+2+3+...) = S
n = 2: (1+2) + (25)(1+2+3+...) = S
or in general, using the formula for the nth triangular number,
(n)(n+1)/2 + [(2n+1)^2](1+2+3+...) = S

Noting 1+2+3+... is just S, and some further simplification:
(n)(n+1)/2 + [(2n+1)^2](S) = S
[n^2 +n]/2 = S - [(2n+1)^2](S)
[n^2 +n]/2 = S(1 - [(2n+1)^2])
[n^2 +n]/[2*(1 - [(2n+1)^2])] = S
[n^2 +n]/[2*(1-[4n^2 +4n +1])] = S
[n^2 +n]/[2*(-4n^2 -4n)] = S
[n^2 +n]/[-8n^2 -8n)] = S

Finally:
-1/8 = S, no matter the choice of n. Very neat and strange.

[historybuff]

You need to prove that S exists in the first place. The fact that it gives the result in these examples proves that it does not. Merely writing S on the board does not prove it exists.

[mitchelvalentino]

This is fantastic. Most Ramanujan videos are confusing or shallow, whereas your Ramanujan content is challenging yet crystal clear. Thank you very much!

[GenericInternetter]

Numberphile: 1+2+3+4... = -1/12 because Ramanujan.

BlackPenRedPen: HOLD MY BEER

[nilsp]

The sum is divergent, but if you consider it in the context of the Riemann Zeta function, the answer should be -1/12. But, of course, just as with 0/0, looking at it from different angles might give you different values, but if not equal, the sum is at least somehow related to -1/12, as seen with the analytic continuation of the zeta function.

[uumlau]

The Riemann/Ramanujan assignment of values to an infinite series works because it establishes a particular pattern of summation. When you group the elements of the sum differently, it's a different pattern and it is easy to get different values (not just -1/8) by grouping things differently. In particular, if you make manipulations that rely on the fact that there are an infinite number of terms and shift them arbitrarily, you'll get all sorts of answers other than infinity or -1/12, mostly because you're essentially adding infinity to one side or the other of the equation.

The "real life" proof of the Riemann/Ramanujan approach is as it applies to the sum of all cubes, which arises in calculating the Casimir effect, the method of derivation yielding a sum of all cubes which is divergent. Using the analytic continuation of the zeta function yields 1/120, which turns out to agree with the experimental answer. Therefore this method of assigning a value to an infinite sum has a real meaning. Remember that mathematicians used to think that sqrt(-1) was mathematically meaningless, too, until it was found to be extremely useful in physics and could be regarded as a rotation of 90 degrees.

The sum of all integers plays a role in String Theory, and that theory uses -1/12 as a valid result, but I don't regard it as the same kind of real life proof, because there are no experiments that can be done to verify it. For me, it is much more interesting when nature itself proves that the math is right.

[omamba]

You can continue the pattern indefinitely and get -1/8 each time. 3 + 25s = s | 6 + 49s = s | 10 + 81s = s

[ZennerVOID]

Srinivasa Ramanujan, Not every legend lives long. He lived BEYOND !!

[jzanimates]

Can you try and find out what sqrt(2) factorial is!

[faisalinc.]

I am seriously concerned about all those -1/12-Videos regarding the infinite sum of all natural fear that some non-math-people start to think that this sum actually HAS a finite value ....

Nice video though, showing that assuming this divergent sum to be finite leads to multiple contradicting solutions...

[ayushgangrade]

It's a divergent series, OK
Now it will focus in negative number line
This series is getting answer as
- 1/12. U are changing the concavity of series by adding them as pairs of 3 and 25 thus focusing the series on - 1/8, it changes the magnitude of infinity by pairing in 3 and 25

[markovnikovchung]

Neither of those, the series diverges