So, I think my mind just broke. Why is the sum of all integers -1/12?

Asked by ragingloli (40259) January 9th, 2014
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The video lost me about 40 seconds in then he made this leap from 1+2+3+4… to 1+ x2 + 2x squared and so on. That may be obvious to a mathematician, but to me it was a giant leap of faith. And logically it doesn’t make sense either.

So I am just as confused as you are.

elbanditoroso (21110)

It is another example of how you can be totally brilliant and not have a fucking clue.

filmfann (43667)

For me, the most important statement in the whole video occurs at about 1:48:

1+x+x^2+x+x^3+...=1/1-x if and only if x>1

Strauss (19055)

How does this not violate the basic algebraic law that positive#+positive#=positive#? My mind is blown as well.

dxs (14330)

This is not a “sum” in the traditional sense. It is called the Ramanujan summation, which is a technique to assign finate sums to infinite divergent series. It is rather somewhere between summation and integration.

PhiNotPi (12611)

So in the traditional sense, the sum of all natural numbers does equal infinity?

dxs (14330)

@dxs, – but suppose you take that sum (infinity) – can you add 1 to it?

elbanditoroso (21110)

The problem is that I treated infinity as a number instead of a concept in that last post. There is no numerical answer to the sum of all natural numbers because the amount of natural numbers is infinite. So it seems like you can’t even apply arithmetic to this. I think that’s the problem.

dxs (14330)

I always loved playing with infinite series.

This seems like a bit of sleight of hand. The tricky part is the use of the formula:
1 + 2x + 3x^2 + ... + n x ^(n-1) + ... = 1/(1-x)^2
This is perfectly legitimate provided -1 < x < 1.
Then the formula was applied to x = -1 to get:
1 -2 +3 -4 +5 – ..... to get 1/1 – (-1))^2 = ¼
This is a divergent series for x = -1 and strictly speaking, there is nothing you can say about it.

For example, (1 -2) + (3 -4) + (5–6) + .... = -1 + -1 + -1 + ... = – infinity
On the other hand:
1 + (-2 + 3) + (-4 +5) + ... = 1 + 1 + 1 +... = + infinity

The math in @PhiNotPi ‘s link is beyond my understanding, but apparently what mathematicians have done is to extend the formulas for convergent series to divergent series and somehow make sense of it.

@LostInParadise Don’t worry, some of it is beyond my understanding also.

PhiNotPi (12611)

Isaac Newton is smiling right about now… this part of calculus, diffferentials has given math students a new way to play with numbers. I worked on this for a while and it does make sense as a brain tease. Actually the key to this is pre Newton (Ada Lovelace, I think) because of raising a number to the power of two. When physicists grapple with quantum mechanics (still an unsolved mystery) they come up with this stuff! So… this is a very clever brain tease.

herculies (645)

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