On 29 Jan 2014, at 22:28, Russell Standish wrote:
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.
Yes. It is Abel summation.
Basically, for 1 - 1 + 1 - 1 + ..., Euler substituted x by 1 in the
geometric identity series:
1/(1+x) = 1 - x + x^2 - x^3 + x^4 - ...
That identity is easy to prove for IxI < 1. It gives the "Abel
summation" for x = 1. Euler *defined* its value for x = 1 by the left
hand side. That definition can be generalized for all series being
some Taylor series with coefficient smaller than 1.
Bruno
Cheers
On Wed, Jan 29, 2014 at 10:11:13PM +0100, Telmo Menezes wrote:
On Wed, Jan 29, 2014 at 9:56 PM, LizR <lizj...@gmail.com> 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.
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