ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12

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The sum of all natural numbers (from 1 to infinity) produces an "astounding" result.

Tony Padilla and Ed Copeland are physicists at the University of Nottingham.

We also hear that Chapter XIII of Konrad Knopp's book, "Theory and Application of Infinite Sequences and Series", is very good if you can get your hands on it.

NUMBERPHILE

Videos by Brady Haran

[jojogothic]

so guys lesson today is if someone offers to give you 1 dollar today 2 dollars tomorrow ect ect dont take the deal since he is obviously trying to steal you

[sempertard]

i always multiply both sides by zero. Seems to fix things up pretty well.

[perseusgeorgiadis]

My IQ increased by -1/12 after watching this

[rupertolababwe]

The analytic continuation of the Riemann Zeta function does indeed map -1 to -1/12, however this does not mean that the sum of all positive integers is -1/12. The whole point of analytic continuation is to extend the function to the domain where the original function is divergent, and after doing that u CANNOT say that the original function maps the analytically continued domain to all these extended points

[bilbo_gamers]

A simple stack overflow bug. God will patch it in the next update.

[viola_case]

Mathematics when YouTube removes the dislike button:

[Prs]

Trolley Problem: A trolley is on a track headed towards one person, and after this one person is two people, and after that is 3 people, and so on. You can flip a lever to send the trolley onto an empty track. Do you flip the lever?

[almircampos]

After having watched this video for infinite times, I realized that my knowledge had increased by a -1/12 factor every time I watched it.

[UnknownRager]

Me before watching this video: liar
Me after watching this video: cheater

[carpaltullar]

We were allowed to make an intuitive conclusion about 1-1+1-1…, but weren’t allowed to make a much more intuitive conclusion about 1+2+3…

[GamingBlake]

It seems like there's all kinds of tricks you can pull to get whatever result you want, once you throw rigor out the window. For example, he took the average of 1 + 1 - 1 + ... to get 1/2. You could also do this:

1 - 1 + 1 - 1 ...
= (1 + 1 + 1 ...) + (-1 - 1 - 1 ...)
= (1 + 1 + 1...) - (1 + 1 + 1...)
= 0

[jiggybau]

Let me prove that 1 = 0, using this premise:

S1 = 1 + 2 + 3 + 4 + 5 ... = - 1/12
S1 - S1 =
1 + 2 + 3 + 4 + 5 + 6 ...
- 1 - 2 - 3 - 4 - 5 ...
= 1 + 1 + 1 + 1 + 1 + 1 ...
Since S1 - S1 = - 1/12 - (- 1/12) = 0
It follows that 1 + 1 + 1 + 1 + 1 .... = 0
Let's name this sequence S2:
S2 = 1 + 1 + 1 + 1 + 1 ... = 0
Now let's subtract it from itself:
S2 - S2 =
1 + 1 + 1 + 1 + 1 ...
- 1 - 1 - 1 - 1 ....
= 1
Given that S2 equals 0, we can also write this as:
0 - 0 = 1
Which implies that 1 = 0.

[stormysamreen]

Tony: "The answer can be either 1 or 0, so we take the average 1/2
Me: "Ok, now that's where you screwed up"

[coach_rohit]

I think the biggest assumption is that S1 is 1/2 which I think is the reason why we got all the natural numbers sum to -1/12

[ludvigpio]

But S1 and S2 are divergent series, they can't be assigned a value. This video just shows that if you try to assign a value to divergent series you can prove nonsense such as sum of all positive numbers equal -1/12

[paulzapodeanu]

To quote a math teacher from my uni: "It's extremely unpleasant to approximate solutions that don't exist."

[shreyansh]

If you're from 11th-12th science, and you got some amazing Professor who sometimes taught you this type of curious and out of the syllabus problem, just to keep you hooked to the wonder of science and Mathematics, you're lucky.

[andrewazariah]

When you do 2S2, isn't it going against rules when you shift the bottom set of numbers by one place ?

[BrianBell]

This is bollox. For S1, you can't just stop on odd or even. Infinity is infinity. It is a concept. It isn't a number. You have to keep going.

[divergentmaths]

A reminder of the golden rules to be adhered to when dealing with divergent series:
1) Do not use brackets.
2) Do not remove any zero (unless you have proven that the divergent series is stable).
3) Do not shuffle around more than a finite number of terms.
Not adhering to these rules yields incorrect sums.