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

On 21 Jun., 23:19, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 21 Jun., 20:08, Virgil <vir...@xxxxxxxxxxx> wrote:> In article <1182430766.206895.121...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
If sum is not considered to be meaning sum, then he should have used
another symbol. Why should there be any sensible meaning in using a
symbol, say "_+ _ + _ + ..." and attaching to it a meaning involving
"_- _ + _ - ..."? Some tactical manoeuvre to cope with the paradoxes
of set theory?

WM is apparently unfamiliar with the common practice of using the "+"
symbol to mean a number of different things simultaneoulsy. For example,
in a vector space, with lower case indicating scalars and upper case
indicating vectors, consider the expression (a + b)(A + B).

When used in the natural numbers, for instance, the symbol "+" is well
defined. It has a fixed meaning and cannot be changed unless clearly
stated. At least in mathematics this is so.

Yes.
The symbol "+" stands for a certain binary operation +: NxN->N,
usually written in infix notation, such that
a + 0 = a
a + successor(b) = successor(a+b)
Wonderfully defined.
Which two numbers were you adding then?



By the way, why don't you define Sum{n in N} (1/2^n) = -5 ?

I have given arguments in a previous post, why certain extensions
of the Sum symbol are useful at different levels.

The only "immediately allowed" application of the capital sigma
is to finite index sets as per recursive definition via "+".

A very practical extension is to define Sum{i in I} a_i
as the sum of all non-zero numbers occurring, provided all but
finitely many a_i are 0.
Note that this is especially fine because we have what can be called
generalized commutativity and associativity:
1) Sum{i in I} a_i = Sum{i in I} a_p(i) whenever p is a bijection I->I
and at least one side is defined
2) Sum(i in J) a_i + Sum{i in K} a_i = Sum{i in I} a_i
whenever I is the disjoint union of J and K and at least
one side is defined.
3) Sum{i in I} a_i + Sum{i in I} b_i = Sum{i in I} (a_i + b_i)
if at least two of the sums are defined.

For certain index sets I (for simplicity, consider only I=N),
it makes sense to investigate partial sums and (given some topology
in the target domain) their convergence.
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 definition that *extends* the meaning of Sigma (fortunately
the limit of the partial sums coincides with the sum as given
above if almost all a_i are zero) to certain sequences where the
sum is not defined.
One *could* try to define it as twice the limit of the partial sums,
but that would conflict with the previous definition for the case
of only finitely nonzero summands.
Therefore, the usual definition makes a lot of sense and has
been used successfully for ages.

Note that 1) and 2) above do not hold in general for this extension!
This is a lesson beginners have to learn thoroughly when being
introduced to series:

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.

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

Still no bad feeling with that?


.

Relevant Pages

  • Re: abundance of irrationals!)
    ... > Having infinitely many finite sums no more creates an infinite sum than ... > having infinitely many natural numbers creates an infinite natural. ... to prove that I S- S I < eps for the partial sums Swhich are ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >>> series is converging. ... >> or axiom that gives meaning to them). ... >> talk about sums, but only about differences between pairs of elements. ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >> series is converging. ... > or axiom that gives meaning to them). ... > talk about sums, but only about differences between pairs of elements. ...
    (sci.math)
  • Re: COUNTIF In This Case?
    ... >bottom that sums the number of occurances where the date is less than ... >Meaning, if there's any that meet this, the result is one, and if not, ...
    (microsoft.public.excel.worksheet.functions)
  • Re: Need help with infinite series
    ... >>>It converges by the alternating series test. ... >>>It sums to approximately 0.6033175. ... take the average of two consecutive partial sums. ... you did we should get 8 decimals right instead ...
    (sci.math)