You are allowed to do divergent sums, sure. But that does not make them equal.
-- Raul On Wed, Apr 13, 2016 at 10:36 AM, R.E. Boss <[email protected]> wrote: > >> From: Chat [mailto:[email protected]] On Behalf Of Raul >> Miller >> Sent: dinsdag 12 april 2016 15:38 >> >> On Tue, Apr 12, 2016 at 6:38 AM, R.E. Boss <[email protected]> wrote: >> > Sorry my answer took a while, but I attended a lecture on "Infinite >> bookkeeping", from which I learned that >> > 1-1+1-1+1-1+1-1+1-1+.....= 1r2 (the dots meaning 'ad infinitum') and >> > 1-2+3-4+5-6+.... = _1r4 , >> > unfortunately this cannot be experienced with J. >> >> Well... https://plus.maths.org/content/infinity-or-just-112 touches on >> this subject. > > > The lecturer and (some of) his audience knew both the site and the youtube it > mentions. > So he pointed out what you are – mathematically – allowed to do with > divergent sums. > Let s= a1+a2+a3+....then allowed are > 1) s=0+a1+a2+a3+.... > 2) k*s= k*a1+ k*a2+ k*a3+.... > 3) if t=b1+b2+b3+... then s+t=(a1+b1)+(a2+b2)+(a3+b3)+... > (the last rule I don't remember right now) > > So you get > s=1-1+1-1+1-1+1-1+1-1+..... > s=0+1-1+1-1+1-1+1-1+1-1+..... > 2s=1, so s=1r2. > > But watch out: what is 1-1+0+1-1+0+1-1+0+....? > Well, same trick > s=1-1+0+1-1+0+1-1+0+.... > s=0+1-1+0+1-1+0+1-1+0+.... > s=0+0+1-1+0+1-1+0+1-1+0+.... > so we get > 3s=1 and thus s=1r3 (!) > > I loved that. > > > R.E. Boss > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
