On 18 Jan 2014, at 02:27, LizR wrote:

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The demonstration that the sum of the positive integers is -1/12relies on the assumption that the sum of1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 -1 .... is 1/2However 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. Orit could be one of the other values mentioned - if it's 0, we getthe 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 everything-list+unsubscr...@googlegroups.com. 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.