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I was just watching this video: http://www.youtube.com/watch?v=w-I6XTVZXww

In it, a professor working at the Nottingham University( Dr. Ed Copeland I think) shows how 1+2+3+4+5....+ ∞ = -1/12 Is this a joke? Or does it contain somewhere within it a flaw?

Now, I know that this channel and the people running it are by profession researchers so it does make me think whether this is true.

but at the same time, you are adding positive numbers on one hand and getting a negative sum?

The video also shows how 1-1+1-1+1-1.... =1/2 and that is counter intuitive too, how can you get a fraction when you are only adding together integers? I do however, painfully accept this as the truth because this is an infinite G.P. with r= -1. and I can get the answer using the formula.

what I am asking is:

  1. Is this actually true or altogether false, or a limitation to the current system of mathematics and accepted as true only because otherwise, mathematics would be proven wrong?

  2. Is there any intuitive explanation to this (both the series that are mentioned), or can I only see it using mathematics?

  3. The video mentions that their are actual uses in physics for this. What are they?

I have read other answers and the opinions in them are conflicting, some say it is false and some say that it is true. Also, other questions don't seem to answer all hree of the questions posed.

Aayush Mahajan
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    Interpreted in the standard way, it is false. However, it hides a deep truth. – Harald Hanche-Olsen Jan 09 '14 at 17:36
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    Whether it is a joke or not is irrelevant; clearly it contains a flaw. See if you can find the flaw. Start by clearly defining what those "..." mean. – Eric Lippert Jan 09 '14 at 17:36
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    The idea underlying these counterintuitive "sums" isn't the--flawed--notion that the limit as $n$ goes to infinity of the sum of positive integers from $1$ to $n$ is, in fact, $-1/12$, but rather that there's a consistent way to assign a real number to certain formal infinite series that allows us to do computations with them even when sums for those series don't exist. – Nick Jan 09 '14 at 17:39
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    This has been asked a few days ago, please look for further answers – user88595 Jan 09 '14 at 17:40
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    There are several ways to "define" the value of a divergent series $\sum_{n \geq 1} a_n$ using complex analysis. One way is to study the power series $f(z) = \sum_{n \geq 1} a_nz^n$ and the other is to study the Dirichlet series $g(s) = \sum_{n \geq 1} a_n/n^s$. Formally, $\sum a_n$ equals $f(1)$ and $g(0)$. Analytically, if $f(z)$ converges for small $z$ and has an analytic continuation to $z = 1$, you could define $f(1)$ to be that value. And if $g(s)$ converges for ${\rm Re}(s)$ large and has an analytic continuation back to $s = 0$, you could define $g(0)$ to be that value. That's all. – KCd Jan 09 '14 at 19:04
  • To make sense of $\sum_{n \geq 1} n$ in this way, consider $\sum_{n \geq 1} nz^n$ and $\sum_{n \geq 1} n/n^s$. The power series converges for $|z| < 1$ and equals $z/(1-z)^2$, which doesn't have an analytic continuation to $z = 1$. The Dirichlet series converges for ${\rm Re}(s) > 2$ and equals $\zeta(s-1)$. It has an analytic continuation to the whole complex plane except at $s = 2$, and its value at $s = 0$ is $-1/12$. This provides a reason to say if we want to assign $\sum_{n \geq 1} n$ a finite value then it could be decided to use $-1/12$ as the value. – KCd Jan 09 '14 at 19:08
  • Harald Hanche-Olsen: I wonder whether we mortals deserve to know that truth.. Is it some kind of duality in math, like quantum-corpuscular duality in physics? – mykhal Jun 21 '15 at 16:23
  • @user88595 This "equation" unfortunately became very common and was mentioned here very often. But no matter in which sense it is valid, it is simply false if it is just stated without the necessary context. – Peter Jun 08 '18 at 07:46
  • @mykhal I do not understand why (mathematical) trues should not be allowed to be known by everyone who is interested. But this kind of "truth" (the meaning of the "equation") need not to be known. – Peter Jun 08 '18 at 07:48

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