I like the Catalan example --- but it doesn't quite address Mike's question. When a sum is assigned to a divergent series via analytical extension, the value obtained will vary according to the analytical representation involved. The original example apparently implicitly interpreted terms as coefficients in a Dirichlet series; utilising Taylor series yields instead 1 x^0 + 2 x^1 + 3 x^2 + ... = 1/(1 - x)^2 at x = 1 , yielding (some flavour of) infinity. Anyway, Jim isn't the pope --- I am. WFL On 3/1/14, James Propp <jamespropp@gmail.com> wrote:
Here's a short path from divergent series summation mumbo-jumbo to total nonsense:
If you sum 1+1+2+5+14+42+... (the Catalan numbers) by plugging x=1 into the formula 1+x+2x^2+5x^3+14x^4+...=(1-sqrt(1-4x))/2, you'll get a non-real complex number a+bi with b non-zero.
Now take the complex conjugate of the equation 1+1+2+5+...=a+bi. The left hand side (being a sum of ordinary integers) stays the same, but the right hand side becomes a-bi.
So a+bi=a-bi, so 2bi=0, so i=0, which gives -1=i^2=0^2=0. Adding 2 to both sides gives 1=2. Therefore (for the usual reason) I am the Pope.
Jim Propp
On Friday, February 28, 2014, Mike Stay <metaweta@gmail.com> wrote:
On Fri, Feb 28, 2014 at 7:03 PM, James Propp <jamespropp@gmail.com<javascript:;>> wrote:
Dan, can you say more about why you object to using the equation 1+2+3+4+...=-1/12 in this way? Is this equation, for you, exactly as objectionable as (say) 1+2+4+8+...=-1, and for the exact same reason?
And is that reason the conviction that "1+2+3+4+..." must (in the absence of side explanations) connote the limit (in Cauchy's sense) of 1+2+...+n as n goes to infinity?
I would guess his objection has to do with abuse of notation, in that one is implicitly assuming a nontrivial summation method. Are there series whose sum gives different results if you use Abel, Borel, Cesaro, Euler, or Lambert summation? If not, then perhaps the objection is unfounded. -- Mike Stay - metaweta@gmail.com <javascript:;> http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com