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

On 22 Jun., 22:49, hagman <goo...@xxxxxxxxxxxxx> wrote:
On 22 Jun., 14:25, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

On 22 Jun., 10:20, hagman <goo...@xxxxxxxxxxxxx> wrote:

This allows one to extend the meaning of the capital sigma
to stand for the limit of these partial sum, provided the limit
exists.
This is NOT a sum.

It is a sum. But that is not the question. *** wrote sum{n in N}n. It
does not matter whether you accept it as a sum or as a limit.

It is not a limit either.
It's a definition.

It is a definition which is in contradiction with the fact that no
definition of this sum in R is *possible*. Otherwise it would already
have been defined.



Things we know about finite sums need not remain valid
if we switch to series.

With that in mind, have a look at your conclusion:

Finite sums of non-negative integers where at least one
summand is positive are never equal to zero.

Everything we know about finite sums also holds
for any usable extension to infinite sums.

Not everything holds, but this thing holds!

Why?
How do you decide which properties should hold and which not?
And why should your decision bind us?

Because there is no other outcome possible.
Either there is an infinite set N, then the sum {n in N} 1 = omega.
Or there is no set N, then every sum {n in N} 1 = finite > 0.



Therefore defining Sum{n in N} n = 0 makes no sense.

Still no bad feeling with that?

I teach this as follows:
If a positive sequence (or positive sequence of partial sums) has no
limit in R, then we can use the reciprocals of this sequence. If the
sequence of reciprocals has limit 0, then the sequence has the
improper limit oo.

This is also a possible and sometimes useful *definition*.

It is the general and only result of summing up this series in
orthodox mathematics. It is not a definition.
Otherwise you could also define that the number of natural numbers is
0 or that
sum {n in N} 1/2^n = 5.

Here, you give up other useful things, e.g. you must have 2*oo=oo
and have no field anymore. Also, the convergence to oo is different
as given epsilon>0, one cannot find M such that ...

There is a definition of epsilon around oo, but this definition is not
required to obtain the only result.

But it remains just that: a definitions (of the symbol sum a_i
in such cases and in a way of also of the symbol oo), and
one *can* define differently.

One cannot define at all, not in mathematics at least.


There is a linear map from a subspace of the vector space of
sequences
to the reals that coincides with the arithmetic sum if almost all
members of the sequence are 0, maps sequences belonging to convergent
series
to the limit AND maps (1,2,3,...) to 0.

This indicates above all, that this sequence cannot
have a limit in r.

Agreed. Which makes sum a_i undefined until one defines it.

It makes it undefined or makes the definition wrong.

If you don't believe in this simple rule, then you may apply the same
idea a bit more formally as the epsilon surrounding of oo. Please
inform yourself about this case, for instance at:http://eom.springer.de/L/l058820.htm

The result is the same. Therefore defining Sum{n in N} n = 0 is
nothing but a perverted result of bad logics, following the insane
"idea":
Sum{n in N} n cannot be defined in R.
Therefore Sum{n in N} n is undefined in R.
So I can define Sum{n in N} n as any r in R.

Or even as something not in R, as you do by saying Sum(n in N)=oo !

Of course. Every definition in R is excluded. The three steps above
are purest nonsense and were intended to show it.

Regards, WM

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