I think the problem is that for non-converging series, there are multiple similar tricks you could do that would give different answers...for example:
S = 1-1+1-1+1-1... -1*S = -1+1-1+1-1+1... For a finite or converging series, the order of the summation doesn't affect the final sum, so if in -1*S we switch the places of the 1st and 2nd term, similarly with the 3rd and 4th, etc., we get: -1*S = 1-1+1-1+1-1... = S But if S = -S this implies 2S = 0 which means S=0. Another way of getting S=0 would just be to group terms together like this: S=1-1+1-1+1-1...=(1-1)+(1-1)+(1-1)+...=0+0+0..., which is a convergent series with a well-defined sum of 0. Likewise, you could also group the terms together like this: S = 1-1+1-1+1...=1+(-1+1)+(-1+1)+(-1+1)+...=1+0+0+0..., which again is a convergent series with a sum of 1. On Wed, Jan 29, 2014 at 3:56 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? > > -- > 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.
