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Does 0+1+(-1)+2+(-2)+...have a convergent sum?
Posted:
Dec 5, 1996 3:02 PM
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I would like to propose that the sum of all reals or integers is
convergent. Yes I realize what I am saying. As well, 0+1+1+... is
convergent toward a real number, but I must redefine mathematics as we
know it, or atleast, reveal, challenge, and destroy our basic assumption
that zero is nothing. You should be skeptical, but explore with me
anyway. It will be more exciting either way.
Last week I asked, Does any branch of mathematics consider zero to be
an infinite number? meaning a whole infinity, because the sum of all
real numbers would equal the greatest number of all, that sum being
zero.
1 + (-1)) + (2 + (-2)) + (3 + (-3)) +...
Peter Verthez wrote:
> This is not correct, it depends on how you add them. For sake of
>simplicity, we'll take the set of [integers] instead of real numbers.
>Then: 0 + (1 + (-1)) + (2 + (-2)) + ... = 0 + 0 + 0 + ... = 0
>But: 0 + 1 + (2 + (-1)) + (3 + (-2)) + ... = 0 + 1 + 1 + ... = oo
>So it's wrong to say that the sum of all whole numbers is zero, rather
>it is undefined.
I objected, and said, or at least tried to say, that the way Peter had
rearranged the two sets caused a remainder at the end of the equation.
I wrote:
At the end of the equation, there will be no positive number to add to
the last negative number. That last negative number happens to be
equal but opposite to 1 + 1 + 1 + 1..., and the sum returns to zero.
Rene Bos read this and asked:
> What is the last element of an infinite sequence?
My answer was essentially that:
The last two elements of an infinite sequence are +oo
and -oo, implying that:
+oo is equal to 1+1+1+...
and -oo is equal to (-1)+(-1)+(-1)+...
As well:
+oo + (-oo) = oo In words: the positive and a negative infinities
are each half of a whole infinity.
And so attempting to solve the equation below:
0 + 1 + (2 + (-1)) + (3 + (-2)) + (4 + (-3)...
I wrote:
...((+oo-2)+(-oo+3))+((+oo-1)+(-oo+2))+((+oo)+(-oo+1))+ (-oo) = 0
same as ...1 + 1 + 1 + (-oo) = 0
Obviously, I treated the set of integers as a whole set rather than an
incomplete set. I was suggesting two new real numbers, a positive
infinity and a negative infinity, as I expect has been considered and
abandoned before.
There were two problems with this solution showed to me by others.
Peter pointed out half of the first difficulty by explaining the rule of
thumb:
> There *always* is a next positive number (or differently stated,
>there is *no* last negative number).
And this is true if counting up from zero or down from +oo or -oo.
In a sense, 0 + 1 + 2 + 3 +... and ... (+oo - 2) + (+oo - 1) + (+oo)
would expand away from each other since they could not ever meet each
other and thus would not relate as a whole set.
Mike Housky explained the problem even more clearly:
>Here's the problem. An infinite series cannot always be rearranged
>without changing the limiting value--if there is one. One condition that
>guarantees that it is safe to rearrange terms is if the absolute value
>of the terms of the series have a convergent sum. The formal term
>for this property is "absolute convergence" and it is obviously not a
>property of the sum of all integers.
>An infamous example of a series that is convergent, but not
>absolutely convergent, is the alternating harmonic series:
> 1 - 1/2 + 1/3 - 1/4 + 1/5...
>Leaving rigor aside, the positive terms add up to +infinity and the
>negative terms add up to -infinity. Adding these two gives who-
>knows-what. However, the series does converge (to ln(2), if you are
>interested) if you add the terms in order.
=================
IF any of the "sum of all integers" equations written above did converge
toward a single last number, then it would be possible to say that the
sum of all integers is defined, rather than undefined. If there was
convergence then we could say there is a final number to an infinite
sequence. Instead, the value of the numbers just keeps getting larger
and larger. Dont they?
IF YOU ARE NOT SITTING DOWN, YOU HAD BETTER FIND A CHAIR.
It was Rene Bos who helped me see more of the big picture to what I
knew very well in other ways, by pointing out that according to my
reasoning, every number would be infinite.
Rene Bos wrote:
> ...according to your reasoning, the set Z/{-1} (in words: the set
>of all integers, except for -1) would have the sum 1, and 1 would be
>infinite.
If the sum of all numbers is the greatest number, then the sum of all
numbers, except (-1), equals one. And this is true and part of what
I have meant all along.
But this is important, it would not be a whole infinity.
What is really important, amazing, incredible...the solution that I am
excitedly trying to communicate, is that when we rightfully treat all
numbers as infinite numbers, then the true value of 1 would be less
than the whole, or less than zero. And 2 is also an infinite number,
and is both less than the whole, and less than 1.
Do not think I am just playing a trick and reversing the idea of value.
It is not just a reversal. I am radically redefining the meaning of
value, and at the heart of the matter, I am challenging our
assumption that zero is nothing. The result of this method, the
implications are mind boggling, and honestly, the most beautiful
mathematical principle unknown to man.
If every number is the sum of all numbers except its opposite, and
so every number is an infinity, then greater numbers have smaller and
smaller values in relation to zero. Stay with me.
If all numbers are infinite, and zero is the only whole infinity, then:
1 is a smaller infinity than zero.
2 is a smaller infinity than one.
10 is a smaller infinity than nine.
1 hundred,
1 thousand,
1 billion
1 trillion, in relation to the true value of smaller numbers, or zero,
have
a smaller and smaller value.
It follows that 1+1+1+1... is a convergent number. Each next sum,
in relation to the whole, has a smaller true value.
It follows from the above, that each consecutive positive number
away from zero is less of an infinity than the preceding number,
because it is that much more less than the whole infinity that is zero.
The same is true of negative numbers.
It follows that +oo and -oo would be the smallest infinities. In extreme
numbers converge to a point. Those points are the absolute smallest
numbers, and yet are half infinities which sum to a whole infinity.
The number line is a spectrum of infinities.
It is exciting in part because it is so opposite to how we now think. We
think of zero as nothing and each number is more. But in truth, when we
realize each number is an infinity, "the really giant line of zeros
around the earth kind of large numbers", are smaller infinities than
5 or 2, and far smaller than the nearly infinite and almost whole
number 1.
How can this be integrated with what we are accustomed to thinking.
No problem. Two apples is certainly more than a single apple. By
recognizing all numbers are fragments of a whole infinity, and
recognizing that the value of greater numbers is less, we are not
disturbing mathematics itself. We are merely challenging and
correcting our assumption that zero is nothing.
The change is profound but very simple. We assume the meaning of
one to be greater than zero, meaning that it is above or greater than
the nothing, the void, which zero represents. I hope this is very clear.
We ASSUME zero is nothing. That is the way we presently
conceptualize zero, and empty space, and that conceptualization
poisons every facet of mathematics and physics; completely blinding us
to the undivided nature of nature.
Two apples is still more apples than one, but for there to be a positive
two apples (matter), there must be a negative two apples (anti-
matter). The two positive apples are less than the whole of the four
apples. For every positive number there is a negative number, and
each is less than the two together. This becomes ever more clear as
we continue and recognize how intimately this method applies to
physics.
I would like to now introduce the term volume into the ideas here.
I have said 1 includes all numbers except (-1), and I will say it
has the largest volume of any positive integer. That volume becomes
increasingly smaller with each next larger integer, and that area,
as does the true value of increasing numbers, becomes smaller and
smaller, converging toward a point.
ALL numbers are infinities. And the two absolute smallest numbers
are +oo and -oo, the numeral volume of which is zero.
Returning to the problem of there always being a next positive (or
negative) number, and how counting up from zero and counting down
from +oo can never join, and would thus expand away from one
another.
Now we recognize the deeper nature of an infinite sequence, as it can
be recognized that
O+1+1+... is convergent while,
(...+oo + (-2)) + (+oo + (-1)) + (+oo) is divergent toward zero,
meaning that they no longer expand away from each other, since the
volume of the numbers increase or decrease.
Beginning from the whole oo + oo -1+ ... counts up and is
convergent toward ...+ (+oo -2) + (+oo -1) + (+oo) which counts
down and is divergent toward oo (zero).
Large numbers actually become infinitesimally small in their value.
1+1+1... is CONVERGENT toward a point. Notice how that point is +oo
which is the smallest positive infinite number, and can only expand up
toward being a whole infinite, sort of like that theory called
spacetime. Of course we must redefine mathematics as we know it to
understand all this, but such is the way of history.
If you do not fully see it yet, the best way to understand it is to
relate this to physics and cosmology. If space is flat it must then
have infinite extension. This is considered fact in any text book.
Flat space is infinite and unbounded as it cannot be bordered.
Relativity aside, a perfectly flat space would inevitably have zero
density.
What then happens to space as the measure of density increases
(positively). The overall volume of the universe becomes smaller; the
volume of space decreases, it contracts and becomes curved, opposite to
the Big Bang, until at infinite density the whole positively dense
universe has become an infinitely small point.
I guess this is the Big Bang theory of numbers. I conclude that both
negative and positive infinities are points, even as numbers, the value
of which cannot become smaller. The two infinities cannot become
less than half of the greater whole. And numbers, the ones we
normally think of as very large, instead become smaller and smaller,
converging toward the point which is +oo OR -oo. With that
convergence we can establish that there is a last negative and
positive real number, and further, no longer treat the set of reals or
integers as an incomplete set.
Rejoice!
Mike Housky wrote:
> 1 - 1/2 + 1/3 - 1/4 + 1/5 ...
>The positive terms add up to +infinity and the negative terms add up
>to -infinity. Adding these two gives who-knows-what. However, the
>series does converge (to ln(2), if you are interested) if you add the
>terms in order.
Obviously, it is important now to distinguish between *a* positive
infinity, and *the* positive infinity, since a descending number is not
*the* positive infinity explained in this paper, and since every positive
number is a positive infinity. Having given some thought to this, I
propose that we name the numbers +oo: Proteus, and -oo: Elea.
Last note: In philosophical terms, we might ask, if zero is a whole
infinity and not nothing, what then has become of the term nothing.
The two smallest numbers, Proteus and Elea can be understood to be
the physical form of something and nothing. Note that the two
absolute simplest (smallest) semantic meanings are something and nothing.
And further, it turns out that there is no such thing as a nothing
without form. A nothing without form, or a non-existence, cannot BE,
and is even a contradiction in terms. The only real nothing is
physical and is equal but opposite to something. The book I am
working on explains much more.
Well, we shall now be able to actually see the infinite universe, of
which we are part and have never really been divided from.
I am honored to be explaining this.
Great joy and
Happy Holidays,
Devin Harris
=====================
http://members.aol.com/spfields1/scitour/contents.htm
Many more ideas at my website; a complete cosmological model,
material from the nearly completed third and final version of my book,
entitled Everything Forever.
Special thanks to Peter Verthez, Mike Housky, and Rene Bos.
Html version of this paper at:
http://members.aol.com/spfields1/essays/math.htm
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