🚀 go-pugleaf

Paulo da Costa writes:
>No. He got his "contradiction" by making an invalid expansion. If you
>really try to do the "long division" the way he did you'll see that the
>remainder after each step contains steadily growing NEGATIVE powers of
>z, so the expansion is not valid in the vicinity of z=0, where he
>proceeded to use it (the expansion converges for z *outside* a radius
>of convergence that includes all zeros of his polynomial -- to prove
>that it converges _anywhere_, he first has to arrive at his
>contradiction...).

I note that Abian has not reposted the analysis taht originally caught
my attention, where the fallacy is equally egregious but much more obvious.

As I recall, he obtained an expression (1 + z + z^2) and "proved" it equal
to zero by synthetic division yielding two "different" results. The obvious
problem with this approach is that it results in 1 + z + z^2 = 0 FOR ALL Z.
The flaw is similar to the one you point out, i.e., one of the series is
divergent. But the result is so ludicrous as to be obviously false at an
extremely elementary level.

mt