# Re: Sum of all natural numbers = -1/12?

```
On 18 Jan 2014, at 02:27, LizR wrote:```
```
```
The demonstration that the sum of the positive integers is -1/12 relies on the assumption that the sum of
```
```
1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 .... is 1/2
```
However that is by no means certain.
```
```
It is 1 - 1 + 1 - 1 + etc.

The non rigorous proof consists in adding

s = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ...

to itself, with a a lift on the right

```
it give s + s = 1 + (1 - 1) + (1 - 1) + (1 - 1) + .... = 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + ... = 1,
```
so 2s = 1, and s = 1/2.

```
That is not rigorous, but you can pursue this up to 1+2+3+4+ ... = -1/12.
```

```
The sum could be undefined, in which case the proof simply fails. Or it could be one of the other values mentioned - if it's 0, we get the sum of the positive ints is either 0 or infinity (because S = 4S). If it's 1, we get the sum of the pos int = -1/6!
```

You allude to that funny proof of that sum. It is not rigorous at all.

```
The reason why 1+2+3+ = -1/12 is "accepted", in some sense, is that the extension of the zeta function (zeta(s) = sum of 1/n^s) on the complex plain is defined everywhere, except on 1 (= 1+0i). And that zeta extension, on -1+0i, i.e. zeta(-1) = -1/12, and looks like 1+2+3+4+5+ ...
```
```
More direct proof are available, and make the proof you allude too, "more rigorous", but they use special criterion of limit, different from the usual one.
```
Bruno

http://iridia.ulb.ac.be/~marchal/

--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.
```