# Thread: Infinite series 1+2+3+4... = -112?

1. ## Infinite series 1+2+3+4... = -112?

I just watched a video of Dr Lawrence Krauss and it was a little hard to hear what he said but he said something like the infinite series of $1+2+3+4+... \infty = -112$ and he says it sounds ridiculous but he would be happy to prove it some time.

Can someone elaborate on this? I don't get what he is saying.

2. ## Re: Infinite series 1+2+3+4... = -112?

Hey uperkurk.

The claim sounds ridiculous to say the least.

There are however cases of alternating series that have different answers (not really, just that they are claimed to), but the key issue to realize about those is that you can't just re-arrange terms and expect to get a proper answer when you have an infinite series.

If a series converges then it has only one answer. If it diverges, then it does not have one specific answer and the final answer does not exist as a fixed value.

If you can point out how he proves this then we can take a closer look at it and find where he is tripping up.

3. ## Re: Infinite series 1+2+3+4... = -112?

Belief in God: Prohibitive or Liberating? Lawrence Krauss & Uthman Badar | ANU, April 2012 - YouTube

I haved linked the time at which he says the infinite series thing.

P.S Please don't freaked out about the title of the video, my question has nothing to do with god or anything.

4. ## Re: Infinite series 1+2+3+4... = -112?

The sum as given is complete nonsense. A necessary condition (though not necessarily a sufficient condition) for a sum to converge is that the terms have to go to 0. The terms in this series clearly do not...

5. ## Re: Infinite series 1+2+3+4... = -112?

So why is Dr Lawrence Krauss making such claims if he is wrong? Did you watch the video Prove It? Maybe I miss heard him.

6. ## Re: Infinite series 1+2+3+4... = -112?

See the Wikipedia article about this series. What Dr. Krauss says is "minus one-twelfth", i.e., -1/12, and not "minus one twelve". Also, thanks for the link to the talk; I'd like to listen to it some time.

7. ## Re: Infinite series 1+2+3+4... = -112?

Originally Posted by uperkurk
So why is Dr Lawrence Krauss making such claims if he is wrong? Did you watch the video Prove It? Maybe I miss heard him.
It was a joke. He was, clearly, showing the danger of applying theorems that only apply to convergent series to series when you have not checked to see ifthey are convergent.

8. ## Re: Infinite series 1+2+3+4... = -112?

Yes, there are many jokes like this.

$\begin{array}{ccccc}\text{Consider: } & S &=& 1 + 3 + 9 + 27 + 81 + \hdots & [1] \\ \text{Multiply by 3:} & 3S &=& \quad \;\;\,3 + 9 + 27 + 81 + \hdots & [2] \end{array}$

$\text{Subtract [2] - [1]:}\;2S \;=\;-1 \quad\Rightarrow\quad S \:=\:-\tfrac{1}{2}$

$\text{Therefore: }\:1+3+9+27+81+\cdots \;=\;-\tfrac{1}{2}$

9. ## Re: Infinite series 1+2+3+4... = -112?

Originally Posted by HallsofIvy
It was a joke. He was, clearly, showing the danger of applying theorems that only apply to convergent series to series when you have not checked to see ifthey are convergent.
It didn't sound like a joke when he said it, considering the nature of the discussion that was going on and the fact he said he can prove it.

10. ## Re: Infinite series 1+2+3+4... = -112?

Originally Posted by uperkurk
It didn't sound like a joke when he said it, considering the nature of the discussion that was going on and the fact he said he can prove it.
There may be a "proof" on the order of Soroban's example, but it's so obviously untrue I can't imagine he doesn't see it as real. I think HallsofIvy nailed it.

-Dan

11. ## Re: Infinite series 1+2+3+4... = -112?

Originally Posted by Soroban

Yes, there are many jokes like this.

$\begin{array}{ccccc}\text{Consider: } & S &=& 1 + 3 + 9 + 27 + 81 + \hdots & [1] \\ \text{Multiply by 3:} & 3S &=& \quad \;\;\,3 + 9 + 27 + 81 + \hdots & [2] \end{array}$

$\text{Subtract [2] - [1]:}\;2S \;=\;-1 \quad\Rightarrow\quad S \:=\:-\tfrac{1}{2}$

$\text{Therefore: }\:1+3+9+27+81+\cdots \;=\;-\tfrac{1}{2}$
That is not a joke. Using the 3-adic completion of the rationals, that infinite sum definitely converges, and does in fact equal $-\frac{1}{2}$. Moreover, $-\frac{1}{2}$ is a 3-adic integer. Its 3-adic expansion is $\ldots 11111._3$ which corresponds to exactly that infinite sum.

12. ## Re: Infinite series 1+2+3+4... = -112?

Yes, it's the same type of joke as 1 + 1 = 0. It's false on regular numbers, but we can define other collections of objects with different operations (e.g., a field of characteristic 2) where this equality becomes true.