As someone pointed out, it requires a non-standard definition of convergence, as these series are non-convergent according to the usual Cauchy definition.
IIRC, it may be Abel summation? I remember Abel summation being mentioned during my elementary analysis course, but nobody seemed to understand what it was. Cheers On Wed, Jan 29, 2014 at 10:11:13PM +0100, Telmo Menezes wrote: > On Wed, Jan 29, 2014 at 9:56 PM, LizR <[email protected]> wrote: > > OK... thanks, I should have guesses it was the zeta function :D > > > > Anyway, I showed this proof to my 15 year old son and he soon put me right > > on why 1-1+1-1+1-1+1... is indeed 1/2. > > > > call the series 1-1+1-1+1... S > > > > then 1-S = 1 - (1-1+1-1+1-1+1...) = 1-1+1-1+1-1... = S > > > > S=1-S, so S=1/2 (which is, I should think, another way of writing Bruno's > > proof, above, but maybe even simpler!) > > > > Actually that does look rigorous. I mean, assuming that infinite series > > exist and can be added up, etc, etc, that answer looks fairly watertight. > > What could possibly go wrong? > > I've noticed something (maybe silly, maybe trivial?). Let's say: > > S(0) = 1 = 1 > S(1) = 1 - 1 = 0 > S(2) = 1 - 1 + 1 = 1 > S(3) = 1 - 1 + 1 - 1 = 0 > S(inf) = 1/2 > > So the summation oscillates between 0 and 1, and at the limit it's in > the middle of these two values. Notices that for > > 2 - 2 + 2 - 2 + 2... > > the summation oscillates between 0 and 2 and it's 1 at the limit, and so on. > > > -- > > You received this message because you are subscribed to the Google Groups > > "Everything List" group. > > To unsubscribe from this group and stop receiving emails from it, send an > > email to [email protected]. > > To post to this group, send email to [email protected]. > > Visit this group at http://groups.google.com/group/everything-list. > > For more options, visit https://groups.google.com/groups/opt_out. > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/groups/opt_out. -- ---------------------------------------------------------------------------- Prof Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders Visiting Professor of Mathematics [email protected] University of New South Wales http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
