

The Ramanujan sum of all positive numbers up to infinity is -1/12 - waterside81
http://kottke.org/14/01/the-sum-of-all-positive-integers

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Zikes
According to the Wikipedia article [1], it equals -1/12 (R), except the R is
written a rather fancy script.

Now, I'm no mathematician, but I'm betting that (R) is important, and affects
the true value of the -1/12, and probably should not be omitted.

[1]
[http://en.wikipedia.org/wiki/Ramanujan_summation](http://en.wikipedia.org/wiki/Ramanujan_summation)

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valtron
There's only one way to add a finite list of numbers, but several ways to
"add" infinite lists. Ramanujan summation is just one way. There are also
other ways to define "addition" on infinite lists: Abel, Borel, Cesaro, Euler,
etc. summation.

They're different from each other because they disagree on certain infinite
sums, but they're still considered "summations" because, for finite lists,
they give the same answer as normal addition.

~~~
Zikes
So the title should instead read "The sum of all positive numbers up to
infinity (for a certain definition of 'sum') is -1/12".

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matmann2001
According to some people...

The sum of alternating +1 and -1 equals 1 if you stop on an odd, and 0 if you
stop on an even. Infinity is treated differently in different fields. I
suppose in physics, it's probably treated as a superposition of even and odd,
so statistically, you could interpret it as 0.5. And I'm sure that works great
for physicists and it fits a number of scientific models. But there are
probably other fields involving mathematics that would find this
interpretation to be silly and not very useful.

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d4mi3n
You're referring to Grandi's Number. Numberphile covered this pretty well on
their youtube channel:
[http://www.youtube.com/watch?v=PCu_BNNI5x4](http://www.youtube.com/watch?v=PCu_BNNI5x4)

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dkhenry
The Ramanujan Summation is not a sum in the traditional sense. So the title is
missing a script R after the -1/12.

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klochner
And a non-linkbait title for the post would be:

 _The Ramanujan sum of all positive numbers up to infinity is -1 /12_

Which is not sensational, exciting, or even that interesting.

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chacham15
I asked this question on the mathematics stack exchange:
[http://math.stackexchange.com/questions/633285/is-the-sum-
of...](http://math.stackexchange.com/questions/633285/is-the-sum-of-all-
natural-numbers-frac112)

It is only true if you redefine what the "sum" of an infinite series is.

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valtron
Achilles runs at 1 unit/s and Tortoise at x < 1 unit/s. Tortoise gets a head
start of 1 unit. How long until Achilles catches up?

Well first, Achilles has to travel 1 unit because Tortoise got a head start.
That takes 1s. By then, Tortoise managed to run x units, so Achilles has to
catch-up, which takes another x seconds, and by then... etc: The time it takes
Achilles to catch up is 1 + x + x^2 + ... Since x < 1, this converges and the
answer is 1/(1-x). It takes 1/(1-x) seconds for Achilles to catch up.

Plot the lines: Achilles runs along A(t) = t, the tortoise T(t) = 1 + x t.
Solving for t: t = 1 + x t = t (1 - x) = 1; t = 1/(1-x).

Ok, now what if x > 1? Let's use x = 2.

1/(1-2) = -1. It takes -1 seconds for Achilles to catch up. Don't believe me?
Plot it: A(t) = t, T(t) = 1 + 2t; Achilles catches up with Tortoise 1 second
_before_ the race starts!

There you have it, a physical interpretation of why 1 + 2 + 4 ... = -1. (I
have no example right now for 1 + 2 + 3 ..., sorry.) So you see, this isn't
nonsense, though I admit it DOES depend on definitions!

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keerthiko
I am fairly certain sum=1/(1-x) holds true only for x<1\. when x>1, it is a
divergent series... and I can't tell if you're joking or not, or I would just
laugh...

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j2kun
This video has been showing up all over the internet, and the math in it is
completely imprecise. Numberphile has lost a lot of respect in my eyes due to
how much misinformation this video has spread.

~~~
jgg
They also created a pretentious video where they explained that the NSA hacked
people's e-mails through a rigged random number generator, complete with an
attempt at explaining elliptic curves and modular arithmetic, that ignores the
fact that barely anyone used that PRNG and most e-mail isn't encrypted
properly in the first place.

Either I'm a really grumpy person, or their videos are actually terrible. I
think there are enough interesting things in math that can be explained
without falsely embellishing uninteresting results and ideas.

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dj-wonk
Unintended consequence: my confidence in string theory just plummeted.

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gregw134
Wikipedia says the sum (1 - 1 + 1 -1 ...) diverges.

[http://en.wikipedia.org/wiki/Grandi's_series](http://en.wikipedia.org/wiki/Grandi's_series)

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deevus
Well yeah it doesn't converge on any particular number so it must diverge.

Like what has already been said I think this is helpful in certain fields, but
bollocks in others.

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jimmaswell
He shifted the series over and added it to itself, which I've seen be called
out as an invalid thing to do on another sum 'proof' with a weird result.

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zem
there was a good discussion of this on /r/math a week ago:

[http://www.reddit.com/r/math/comments/1usu93/1_2_3_4_5_112_n...](http://www.reddit.com/r/math/comments/1usu93/1_2_3_4_5_112_numberphile/)

~~~
dmunoz
I was visiting this comment thread with the intention to post that link if
nobody else had already. For all the hate reddit gets, I love it for content
such as the discussion contained in the above.

Reddit has major issues due to its user base, but one thing I do is browse
certain subreddits once a week (month) by filtering by top posts in the past
week (month). The day by day drone in most subreddits is tiring, but quality
threads hit the top posts.

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pazimzadeh
If 1+1-1+1-1+1... can be approximated to 0.5, then
1000+1000-1000+1000-1000+1000... must be approximately 500, but shouldn't
1+1-1+1-1+1...be equivalent to 1000+1000-1000+1000-1000+1000...?

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kibibu
Note that 1+1-1+1 doesn't fluctuate between positive and negative, it
fluctuates between 0,1,0,1,0,1.

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PopePompus
What is presented here is actually a reductio ad absurdum proof that 1 - 1 + 1
-1 ... does not equal 1/2 (or any other number).

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dj-wonk
Lose 20 pounds -- and your sanity -- using this one weird trick!

I suspect this kind of reasoning is useful in some fields for some particular
kinds of infinite sums, but for the particular summation problem as stated, as
most people understand it, this is rubbish.

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vinceguidry
You should watch the linked video. The logic isn't that hard to follow, and
finds real-world application all over the place.

~~~
dj-wonk
I was using humor there. Yes, I am open to this kind of thinking being useful.
I did watch the video. I followed the logic. (That logic is contradictory from
other perspectives -- see [http://www.quora.com/Mathematics/Theoretically-
speaking-how-...](http://www.quora.com/Mathematics/Theoretically-speaking-how-
can-the-sum-of-all-positive-integers-be-1-12)).

Summary: as most people understand it, the sum of all natural numbers is (a)
positive (b) very large (c) not finite (d) certainly not -1/12.

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smegel
Did Euler overflow his abacus or something?

