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

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  • Subject: [mg132246] Re: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?
  • From: Andrzej Kozlowski <akozlowski at gmail.com>
  • Date: Wed, 22 Jan 2014 03:32:06 -0500 (EST)
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  • References: <20140119075641.AB76D69EE@smc.vnet.net> <20140120090105.33E5969EA@smc.vnet.net> <20140121080228.9BF4B69D9@smc.vnet.net> <8E65D7C9-8567-4032-B629-46FD621486DF@math.umass.edu>



On 21 Jan 2014, at 19:58, Murray Eisenberg <murray at math.umass.edu> wrote:

> Andrzej,
>
> Drat, I tried each documented value for the Regularization option except that one!

Yes, the name Dirichlet for this summation (or regularization) method seems to me non-standard but it was the only one that suggested a relation with the zeta function.
Hardy in Divergent Series called this summation method  Ramanujan summation, since Ramanujan used it all the time and obtained lots of formulas with it,  although the classic equality in this subject of this thread goes back to Euler.

Andrzej


>
> On Jan 21, 2014, at 3:02 AM, Andrzej Kozlowski <akozlowski at gmail.com> =
wrote:
>
>> 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
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>
>>
>
> =97=97
> 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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