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Re: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg132226] Re: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?
*From*: Murray Eisenberg <murray at math.umass.edu>
*Date*: Mon, 20 Jan 2014 04:01:05 -0500 (EST)
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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=92s 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
>
=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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