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Re: How to show 1+2+3+ ... = -1/12 using

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  • Subject: [mg132241] Re: How to show 1+2+3+ ... = -1/12 using
  • From: Itai Seggev <itais at wolfram.com>
  • Date: Tue, 21 Jan 2014 03:49:15 -0500 (EST)
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On Mon, Jan 20, 2014 at 04:00:45AM -0500, Richard Fateman wrote:
> On 1/18/2014 11:52 PM, Matthias Bode 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
> >
> 
> Sure. Piece of cake.
> Sum[a^n,{n,0,Infinity}]   results in  1/(1-a).
> 
> %/. a->-1  tells you that this Sum is 1/2
> 
> Starting from this lie  (the sum is actually divergent), you should
> be able to prove lots and lots of things.

Or more usefully:

In[43]:= Sum[1/n^s, {n, 1, \[Infinity]}] /. s -> -1

Out[43]= -(1/12)

Which is perefectly valid if you understand what analytic continuation is. 

--
Itai Seggev
Mathematica Algorithms R&D
217-398-0700 



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