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Topic: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?
Replies: 4   Last Post: Jan 21, 2014 2:58 AM

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 Andrzej Kozlowski Posts: 226 Registered: 1/29/05
Re: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?
Posted: Jan 21, 2014 2:58 AM

Note that:

In[25]:= Sum[n, {n, 1, Infinity}, Regularization -> "Dirichlet"]

Out[25]= -(1/12)

This is of course, perfectly correct ;-)

Andrzej

On 20 Jan 2014, at 10:01, Murray Eisenberg <murray@math.umass.edu> wrote:

> You may try the Regularization option for Sum, but it doesn't seem to give any finite result for that divergent series.
>
> On the other hand, the video to which you refer relies ultimately upon using Ces=E0ro-summability of 1 - 1 + 1 - 1 _ . . . , which you may implement in Mathematica as:
>
> Sum[(-1)^n, {n, 0, \[Infinity]}, Regularization -> =93Cesaro"]
> (* 1/2 *)
>
> [The video to which you refer is disingenuous in not saying up-front that it's not using ordinary summability but some other form(s) of summability. (The merest hint is a brief glimpse of a page of a text on String Theory where the formula
> 1 + 2 + 3 + . . . = -1/12 is displayed just below a line referring to renormalization.)
>
> As it stands, that video, in my mind, is deleterious to understanding of the mathematics of infinite series destructive of trust in mathematics: it manipulates divergent series as if they were convergent.]
>
>
> On Jan 19, 2014, at 2:56 AM, Matthias Bode <lvsaba@hotmail.com> wrote:
>

>>
>> Hola,
>>
>> I came across this video (supported by the Mathematical Sciences Research Institute* in Berkeley, California):
>>
>> http://www.numberphile.com/videos/analytical_continuation1.html
>>
>> Could the method shown in this video be replicated using Mathematica symbols such as Sum[] &c.?
>>
>> Best regards,
>>
>> MATTHIAS BODES 17.36398=B0, W 66.21816=B0,2'590 m. AMSL.
>>
>> *) http://www.msri.org/web/msri
>>

>
> Murray Eisenberg murray@math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower phone 240 246-7240 (H)
> University of Massachusetts
> 710 North Pleasant Street
> Amherst, MA 01003-9305
>
>
>
>
>
>
>

Date Subject Author
1/20/14 Murray Eisenberg
1/21/14 Andrzej Kozlowski
1/21/14 Andrzej Kozlowski