Re: sum R

On Thu, 17 Apr 2008 12:03:39 EDT, amy666 <tommy1729@xxxxxxxxxxx>
wrote:

On Apr 17, 8:40 am, David C. Ullrich
<dullr...@xxxxxxxxxxx> wrote:
On Wed, 16 Apr 2008 07:23:39 -0700 (PDT), The
Qurqirish Dragon

<qurqiri...@xxxxxxx> wrote:
On Apr 16, 7:59 am, amy666 <tommy1...@xxxxxxxxxxx>
wrote:
taking the sum over the reals in a real
interval.

i would like to read more about it on the
internet.

in general for the interval (0,1) with bounded
range (-1,1) the sum over the reals converges if the
integral over (0,1) is 0.

That may be a necessary condition (I haven't
checked it carefully
enough to be certain), but not sufficient.
Consider the characteristic
function on the rationals, for example. For that
matter, any f which
is 0 almost everywhere can be used to make the sum
whatever you want -
convergent, conditionally convergent, or
divergent. Although to be
conditionally convergent or divergent, this type
of f needs to be
non-0 on an infinite set.

No. Since there's no order given for the terms
we're adding, the
sum cannot possible converge unless it converges
absolutely.
In which case f vanishes except on a _countable_
set.
David C. Ullrich

I assumed that the order of addition of terms was
that a_x is an
earlier term than a_y if x < y. (please excuse the
abuse of notation,
as I don't know if this is appropriate for a possibly
uncountable
sum). That's why I left the possibility of a
conditional convergence.
As for countable, in this context I believe my
stipulation that f is 0
almost everywhere covered that.

your assumption works fine in ZFC.

my sum is considered as a sort of average.

so +1 -1 +1 -1 ... = 0

PS. Yes, there _are_ summability methods that give 0
for that sum. There are also summability methods that
give 42 for the very same sum. Which is why you need to
be much more specific about what you mean.

my highest regards

tommy1729

David C. Ullrich
.

Relevant Pages

  • Re: sum R
    ... Qurqirish Dragon ... stipulation that f is 0 ... your assumption works fine in ZFC. ... my sum is considered as a sort of average. ...
    (sci.math)
  • Re: sum R
    ... Qurqirish Dragon ... your assumption works fine in ZFC. ... my sum is considered as a sort of average. ...
    (sci.math)
  • Re: sum R
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