Re: ** says: Definition: sum{i in N} i = 0

On 6 Jul., 21:56, Virgil <vir...@xxxxxxxxxxx> wrote:


> If you can't see the facts supporting my discovery (lim_[x --> oo]
> P(x)/K(x) in the binary tree) then further discussion is meaningless.

Yes, it is meaningless because you do not see that that limit is *not*
the necessary value.

The limit is not necessary the value, but for continuous functions it
is. lim [x-->o] sinx/x = 1 unless you define another value at x = 0
and by that make the function discontinuous.

WRONG! The function defined merely by f(x) = sin(x)/x is not even
defined at x = 0 unless an addition to that definition is appended to
extend the definition to cover x = 0.

This definition has been given in mathematics once and for all by
l'Hospital.
At least in standard mathematics.


The paths of the tree are
continuous, however.

That is an entirely different form of 'continuity' than the continuity
of a real function at a real point. And neither type holds "at oo".

But this kind of continuity guarantees that the limit of K(x)/P(x) is
the only reasonable value for the consideration of the whole tree,
like l'Hospital delivers the only reasonable value for sinx/x at x =
0.

Therefore the limit P(x)/K(x) is the only
possible choice.

Non-existence of a value "at oo" is the standard choice unless one does
some form of compactification to allow even the possibility of
continuity "at oo" to be considered.

You should look up a good math textbook. There you will find strict
divergence, improper limits, and related stuff.

Regards, WM

.