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Re: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?

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  • Subject: [mg132233] Re: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?
  • From: Andrzej Kozlowski <akozlowski at gmail.com>
  • Date: Tue, 21 Jan 2014 03:02:28 -0500 (EST)
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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 at 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 at 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 at 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
>
>
>
>
>
>
>




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