When you group a series you create a new series. If the original series is convergent, all its groupings are convergent too with the same sum (this is a theorem in my textbook). But if you have a divergent series you can get either one, convergent or divergent. The beauty of this if after grouping you got divergent series, the original series is also divergent (since were it convergent...), which is how it turns out Oresme proof is incorrect, since it does the opposite: it groups the divergent construct, but such a grouping can be divergent. That is why the correct proof of harmonic series divergence is by showing sequence of its partial sums is fundamental, and all pf those are divergent.

This is different from rearrangments per Riemann theorem. There only absolute convergence is immune to rearrangment (of course it also means all [same sign terms] convergent series are immune, since those are all absolutely convergent by default). It can be proved without Riemann theorem though and is in fact proved in my textbook.

Actually, I rechecked Oresme proof, no, that is what is used in Oresme proof, series is proved to be divergent by its grouping. Sorry.

On Saturday, July 2, 2022 at 7:59:37 PM UTC-7, Валерий Заподовников wrote:
> When you group a series you create a new series. If the original series is convergent, all its groupings are convergent too with the same sum (this is a theorem in my textbook). But if you have a divergent series you can get either one, convergent or divergent. The beauty of this if after grouping you got divergent series, the original series is also divergent (since were it convergent...), which is how it turns out Oresme proof is incorrect, since it does the opposite: it groups the divergent construct, but such a grouping can be divergent. That is why the correct proof of harmonic series divergence is by showing sequence of its partial sums is fundamental, and all pf those are divergent.
>
> This is different from rearrangments per Riemann theorem. There only absolute convergence is immune to rearrangment (of course it also means all [same sign terms] convergent series are immune, since those are all absolutely convergent by default). It can be proved without Riemann theorem though and is in fact proved in my textbook.

Telescoping is under parameterization. So, two series are already
related, that they both can have their terms rearranged, this it
results their difference is the same, not necessarily either limit
just telescoping terms, only when their rearranged is shared
with their infinite index, what makes the terms of the series(es).

Also rearrange-ing terms, makes from a convergent series what
includes series that are divergent, or, convergent to something else.

There are two kinds of divergent, off (or hysteresis) and out.
This is that there are divergent series that are unarranged
and "divergent" series that are infinities.

Here when I say "infinities" these are the "infinities of
convergence, the divergence, to infinity", "rail numbers".

It's surprising I have to make up "rail numbers" to talk about
convergence, series, and limits in numbers with infinity.

But, anybody could implement same criterion.

I think a lot of series re-arrangements are only coincidentally
justified, "approximations have an error term anyways, ...",
or for exampling relying on systems where "the re-arranged
cancellative in terms happens to cancel series that telescope
together", that it results "according to scale", happenstance.

Convergence criteria are in limits, series, ....

Series co-scoping co-convergence criteria are in series, terms, limits, .....

Continuity and convergence, series as their terms, make for
that because "digital convergence and really only a few terms
of series besides necessarily and all the terms for algebra",
that continuous convergence and digital convergence, is for
providing what co-convergent series that can be rearranged
together before they are evaluated their approximation,
and also maintaining a bound in the error term.

The "classical digital methods of approximation", are defined
about the same way: meaning as series terms in effect or these
are real numbers, whether the method is both just or convergent,
or just and also just with another, maintaining the convergence
criterion, for methods that are significant in their error bounds
together in the same terms before and after being re-arranged.

Then "adding and subtracting series", also results "and their
are only some few abstract series that keep a limit and others
that diverge one way or the other, and even those that lose
a limit and diverge in the sense of an error bound it was maintaining".

So, all the series are in the same space, then besides that there
are ways to define for the geometric series, ..., basically what
are significant terms the significance is maintaining initial
terms together here bounded under orders of magnitude.

But, after the binary geometric series, then there is out besides
orders of magnitude also orders orders in the geometric scale,
where, "there is 1 plus 1/2 + 1/4 + 1/8 and all the binary geometric
series inverse powers of two, it sums to 2" and "telescoping terms
that don't keep the terms arranged to the binary geometric series
both ways lose its maintainance of convergence criteria".

Otherwise it's quite fraught and difficult, and one of the first
ideas in something like Euler and "let me draw out these terms
that Euler already drew out in telescoping, bad idea", makes for
that like anything else its only an operator framework when it's
defined what why it holds.

This is where of course, "if any series were rendered to its part
and all...... summed together only its own same parts", it's
expected that the _result_ would be the same, where it is for
"any finite series". Of course there's no way to effectively
"reverse the series and start from the infinite side little end",
that permuting terms does, for finite series. Yet, logically
there's that "this doubling term, whatever it fills, is in orders
of magnitude".

The positional then as orders of magnitude, then is keeping
the positional here in convergence (of what are series).

This is only knowing two words, "convergence" and "criteria".
"Grouping of terms" (and re-arranging them) is not usually "allowed", no.
There only exists already what is "allowed", also that it is an operator framework.
The convergence criteria only all say what converges and goes.