4/1 - 4/3 + 4/5 - 4/7 + ... = pi

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4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby phobos rising » Thu Jun 29, 2017 12:13 pm

Reference:

http://marilynvossavant.com/forum/viewtopic.php?f=5&t=2000

_____________________________________________

4/1 - 4/3 + 4/5 - 4/7 + ... = pi is a correct formula, or alternatively,

1 - 1/3 + 1/5 - 1/7 + ... = (pi)/4 is similarly a correct formula.

- - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - -

In regards to the first formula, at any stage where the partial sum involves an odd number of
terms, there is an overage as it relates to the actual value of pi. At any stage where the partial
sum involves an even number of terms, the value is always short of the value of pi.

The value of pi lies somewhere between consecutive terms.

If a finite number of terms in this series were to be added, no matter how large that number is,
the sum would be rational. And pi is known to be irrational. But, that is not the case here.

But, as in the series for 1/3, we are not adding a finite number of terms. We are taking the
limit of a sum. For the limit of the sum of this series made up of fractions, it turns out that
the limit is irrational.
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Re: 4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby robert 46 » Thu Jun 29, 2017 4:03 pm

phobos rising wrote:4/1 - 4/3 + 4/5 - 4/7 + ... = pi is a correct formula...

This is comparable to:
horse-like_animal + single_spiral_horn_protruding_from_forehead + lion-like_tail = unicorn

As you can see, the mere definition of an identifier does not establish existence of anything more significant than that an identifier has been defined. In particular, there is no establishment of the existence of an animal corresponding to the definition of a "unicorn".

In regards to the... formula, at any stage where the partial sum involves an odd number of terms, there is an overage as it relates to the actual value of pi. At any stage where the partial sum involves an even number of terms, the value is always short of the value of pi.

Pi is the definition of the result of an infinite process. As such it manifests the reification fallacy. There is no reason to believe that an endless process returns a final value because there is no finality to an endless process.
The value of pi lies somewhere between consecutive terms.

It lies between consecutive partial sums- all of which are rational numbers.
If a finite number of terms in this series were to be added, no matter how large that number is, the sum would be rational.

It is proved that the sum of any two rational terms is rational. Therefore all intermediate partial sums are rational. To claim that a fictitious result is irrational is merely to establish that the fictitious result cannot exist.
...as in the series for 1/3, we are not adding a finite number of terms. We are taking the
limit of a sum. For the limit of the sum of this series made up of fractions, it turns out that
the limit is irrational.

The limit for the named item "pi" does not exist. In the case of 1/3=0.(3), we see that the left and right sides are tautologous- an identity:
10*1/3=10*0.(3)
10/3=3+0.(3)
10/3-1/3=3+0.(3)-0.(3)
9/3=3
3=3

A) 10*a-a=10*b-b iff a=b
(10-1)*a=(10-1)*b
9*a=9*b
a=b

It has not been proved that a=b. Rather, the equality on the left of A) is true only by the definition that a is identical to b.
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Re: 4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby phobos rising » Sat Jul 01, 2017 1:42 pm

robert 46 wrote:
phobos rising wrote:4/1 - 4/3 + 4/5 - 4/7 + ... = pi is a correct formula...



As you can see, the mere definition of an identifier does not establish existence of anything more significant than that an identifier has been defined. In particular, there is no establishment of the existence of an animal corresponding to the definition of a "unicorn".

I stated a fact at the beginning. robert 46, you have shown you are well ignorant of a lot
of mathematics, especially of series. And in other threads, you have presented dozens of falsehoods, not typos by the way. You're a mathematical crank.

In regards to the... formula, at any stage where the partial sum involves an odd number of terms, there is an overage as it relates to the actual value of pi. At any stage where the partial sum involves an even number of terms, the value is always short of the value of pi.

Pi is the definition of the result of an infinite process. As such it manifests the reification fallacy. There is no reason to believe that an endless process returns a final value because there is no finality to an endless process.

This doesn't make sense. It depends. If you were knowledgable enough about limits, you
would not make that statement, or, you're just being contrary.


The value of pi lies somewhere between consecutive terms.

It lies between consecutive partial sums- all of which are rational numbers.
So, that has nothing to do with its series keeping it from being irrational.

If a finite number of terms in this series were to be added, no matter how large that number is, the sum would be rational.


It is proved that the sum of any two rational terms is rational. Therefore all intermediate partial sums are rational.

Don't change the subject. I explained the value is in between the terms.

To claim that a fictitious result is irrational is merely to establish that the fictitious result cannot exist.

Wrong.

...as in the series for 1/3, we are not adding a finite number of terms. We are taking the
limit of a sum. For the limit of the sum of this series made up of fractions, it turns out that
the limit is irrational.


The limit for the named item "pi" does not exist.Yes, it does.

In the case of 1/3=0.(3), we see that the left and right sides are tautologous- an identity:
10*1/3=10*0.(3)
10/3=3+0.(3)
10/3-1/3=3+0.(3)-0.(3)
9/3=3
3=3

A) 10*a-a=10*b-b iff a=b
(10-1)*a=(10-1)*b
9*a=9*b
a=b

It has not been proved that a=b. Rather, the equality on the left of A) is true only by the definition that a is identical to b.


As a confirmed mathematical crank that you are, robert 46, who is so ignorant of much
pertinent mathematics here, including actual or apparent paradoxes, if you cannot adapt and learn what the facts the are presented, you will be ridiculed. I will not have the patience of JeffJo to let you re-argue as a crank about the same points repeatedly.
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Re: 4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby JeffJo » Sat Jul 01, 2017 1:55 pm

phobos rising wrote:Reference:

http://marilynvossavant.com/forum/viewtopic.php?f=5&t=2000

_____________________________________________

4/1 - 4/3 + 4/5 - 4/7 + ... = pi is a correct formula, or alternatively,

No, it technically isn't. It's shorthand for the correct:
lim(n->inf, a_n=4/1-4/3+4/5-4/7+...+4*(-1)^2/(2n+1)) = pi.

The difference is important, because ignorant people will confuse conclusions made about limits, like this, with others made about properties deduced by induction. Like the proof that 0.(3)=1/3, or like every one of these intermediate sums being a rational number but the limit can be irrational. Another example of that, is that it can be proven that

lim(n->inf, 2-2/2+6/8-20/32+...+2*combin(2*n,n)/(-4)^n)

... is the square root of 2, yet it can also be proven that no rational number can be the square root of 2.

In regards to the first formula, at any stage where the partial sum involves an odd number of terms, there is an overage as it relates to the actual value of pi. At any stage where the partial sum involves an even number of terms, the value is always short of the value of pi.
While true, that does not prove that it converges.

+++++

Speaking of the devil,

Pi is the definition of the result of an infinite process.
I'll assume you meant "the definition of pi is the result..." No, it isn't It is the limit of a finite process, as the process increases without bound.

The limit for the named item "pi" does not exist.
The limit of the partial sums most certainly does exist. It is proven, for an arbitrarily small positive real value d, that there is an integer n such that |pi-a_m|<d for all m>n.

In the case of 1/3=0.(3), we see that the left and right sides are tautologous- an identity:
10*1/3=10*0.(3)
10/3=3+0.(3)
10/3-1/3=3+0.(3)-0.(3)
9/3=3
3=3

A) 10*a-a=10*b-b iff a=b
(10-1)*a=(10-1)*b
9*a=9*b
a=b

It has not been proved that a=b. Rather, the equality on the left of A) is true only by the definition that a is identical to b.
The only thing wrong here, is your intentional insertion of the tautology where it doesn't exist, but you want to find it.

It is your definition of 0.(3) that makes this equality solvable for the unknown (and so not tautologous) X.

X = 0.(3)
10*X = 3.(3)
10*X-X = 3.(3) - 0.(3)
9*X=3
X=1/3.

If you accept that your notation 0.(3) is a valid definition, then it allows the subtraction of the infinite number of terms without resorting (as above) to a limit. If you don't, why do you use it, or care about others who use similar definitions?
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Re: 4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby Gofer » Sun Jul 02, 2017 11:28 am

JeffJo wrote:
robert46 wrote:The limit for the named item "pi" does not exist.


The limit of the partial sums most certainly does exist. It is proven, for an arbitrarily small positive real value d, that there is an integer n such that |pi-a_m|<d for all m>n.


That, of course, depends on the meaning of "exist"! When you do the operation "pi-a_m", you have implicitly presumed the existence of a field called R, the set of all reals. So obviously, "it is [not] proven" by that, but is only the definition of what it means to have a limit.
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Re: 4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby robert 46 » Sun Jul 02, 2017 12:27 pm

@ phobos rising
Do you have the intellect to edit your post to correct the quoting errors???
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Re: 4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby robert 46 » Mon Jul 03, 2017 7:14 am

JeffJo wrote:
robert 46 wrote:Pi is the definition of the result of an infinite process.
I'll assume you meant "the definition of pi is the result..." No, it isn't It is the limit of a finite process, as the process increases without bound.

However, there is no "limit". You can talk about "convergence", but that doesn't mean that what is assumed to be converged to actually exists. We can assume that there is a Land of Oz, and that tornadoes have the propensity to drop houses there on top of witches; but this doesn't mean that the Land of Oz actually exists, and that tornadoes in fact drop houses at any convenient location: i.e. houses never actually are wafted to the Land of Oz. So, the intermediate results are always rational numbers- there is no reason to believe the irrational number is obtained.
The limit for the named item "pi" does not exist.
The limit of the partial sums most certainly does exist. It is proven, for an arbitrarily small positive real value d, that there is an integer n such that |pi-a_m|<d for all m>n.

Yet there is always a d arbitrarily close to the defined pi, without actually getting pi; i.e. where d=0.
In the case of 1/3=0.(3), we see that the left and right sides are tautologous- an identity:
10*1/3=10*0.(3)
10/3=3+0.(3)
10/3-1/3=3+0.(3)-0.(3)
9/3=3
3=3

A) 10*a-a=10*b-b iff a=b
(10-1)*a=(10-1)*b
9*a=9*b
a=b

It has not been proved that a=b. Rather, the equality on the left of A) is true only by the definition that a is identical to b.
The only thing wrong here, is your intentional insertion of the tautology where it doesn't exist, but you want to find it.

No. The tautology is actual: the mathematicians define 1/3=0.(3): perhaps not recognizing that they have introduced a fantastical concept.
It is your definition of 0.(3) that makes this equality solvable for the unknown (and so not tautologous) X.

X = 0.(3)
10*X = 3.(3)
10*X-X = 3.(3) - 0.(3)
9*X=3
X=1/3.

But without the fantastical idea that 0.(3) actually is identical to 1/3, the derivation is meaningless. The mathematicians seem to believe that 0.(3) means something more significant than that it is an identifier equivalent to 1/3: that there is actually an infinite extension of 3s to the right of the d.p. This is like saying that one can in principle get to the end of the universe; but where getting to the end of the universe is nonsensical considering that the universe is logically either endless or finite but unbounded- there is no edge. In practice, getting to an edge is not possible; so it is not sensible to say that it can be done "in principle".
If you accept that your notation 0.(3) is a valid definition, then it allows the subtraction of the infinite number of terms without resorting (as above) to a limit. If you don't, why do you use it, or care about others who use similar definitions?

Because: Introducing 0.(3) as a concept is to introduce a fantastical idea which is not logically tenable. This is how religions start, people introduce fantastical ideas which no one with any sense would accept, and then proceed to build up the fantasy to a degree so colorfully intricate that many foolish people believe it is real. E.g: Sherlock Holmes was England's greatest consulting detective. When in London many years ago my father and I had dinner at a restaurant with a Holmesian motif. It was quaint, but no sane person would believe that the deerstalker hat was actually possessed by Sherlock himself.

*****
Gofer wrote:
JeffJo wrote:
robert46 wrote:The limit for the named item "pi" does not exist.


The limit of the partial sums most certainly does exist. It is proven, for an arbitrarily small positive real value d, that there is an integer n such that |pi-a_m|<d for all m>n.


That, of course, depends on the meaning of "exist"! When you do the operation "pi-a_m", you have implicitly presumed the existence of a field called R, the set of all reals. So obviously, "it is [not] proven" by that, but is only the definition of what it means to have a limit.

Very good. Without the fantastical idea that there is a field of reals consisting of rationals and irrationals, there would be nothing to ponder. To my way of seeing it, convergence is not "to" anything if it is not a rational number. The irrational numbers are like converging to the Land of Oz by the tornado process- pure fantasy.
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Re: 4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby JeffJo » Mon Jul 03, 2017 8:09 am

robert 46 wrote:However, there is no "limit". You can talk about "convergence", but that doesn't mean that what is assumed to be converged to actually exists.
And once again, you insist that others must adhere to your definitions, which you refuse to provide. All while denying that definitions mean anything:
robert 46 wrote:As you can see, the mere definition of an identifier does not establish existence of anything more significant than that an identifier has been defined.
Get this straight: NOTHING "exists" in mathematics. "Exists" is a term you use for what follows from the (inconsistent) axioms and other definitions you choose to accept. There are other, consistent sets. In fact, that is the point of mathematics; not to model what robert 46 thinks is "real" or "actually exists," but to provide consistency.

And the fact that you can only reply to these facts with hyperbole and posturing proves that you can't refute them, and know that you can't.

We can assume that there is a Land of Oz, ...
Irrelevant hyperbole.

So, the intermediate results are always rational numbers...
But what we are interested in, is finding out what the ultimate result is.
... there is no reason to believe the irrational number is obtained.
This is a completely unsubstantiated assertion - a non sequitur.

A number we call "pi" must exist in a consistent mathematics that defines circles. From that,we can derive expressions - both finite and infinite - that must include it as a real number. If it "doesn't actually exists" in what you want mathematics to be, that mathematics is inconsistent. Which is why you refuse to describe it.

The need for a real number whose square is 2 can also be demonstrated, along with the fact that it can't be a rational number. So can "e." And cos(30°). Etc. Etc.

Yet there is always a d arbitrarily close to the defined pi, without actually getting pi; i.e. where d=0.
Um, your point is.... ?

Mine is that d can be arbitrarily small, which is the definition of convergence. And that whatever is converged to has to be considered a number.

No. The tautology is actual:
No, the tautology exists only in your mis-representation. In what you claim is a proof, but is only a redundancy. I showed you the actual proof, which you declined to address. Why is that?

the mathematicians define 1/3=0.(3)
No, they really don't. See the proof you ignored.

But without the fantastical idea that 0.(3) actually is identical to 1/3,
The only assumption is that you can have a string of infinite 3's that you represent by (3). If you want to deny that assumption, stop using it.

From that assumption, it is proven that the decimal representation 0.(3) is the real number 1/3, just like it is proven that the real number represented by the infinite sequence 4-4/3+4/5-4/7... has to be what is defined elsewhere to be, and is necessary to exist, real number we call "pi." You can get arbitrarily close to this value proven by increasing the length of the sequence.

The mathematicians seem to believe that 0.(3) means something more significant than that it is an identifier equivalent to 1/3
Certainly it means something "more significant." But it is not an "identifier." Mathematicians recognize that what you misrepresent is a representation of an arithmetic sequence that converges to 1/3, not that it is an "identifier" - please, define what you mean by that if you can - of any form.

Too bad you refuse to. It shows that you are the crank phobos accuses you of being.

To my way of seeing it, convergence is not "to" anything if it is not a rational number. The irrational numbers are like converging to the Land of Oz by the tornado process- pure fantasy.
And yet you STILL cannot define what it means to exist - only that you don't like it if somebody else claims things you don't like do exist.

And you ignore that there are other ways to prove that irrational numbers must exist, besides convergence.
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Re: 4/1 - 4/3 + 4/5 - 4/7 + ... = pi

Postby phobos rising » Thu Jul 06, 2017 3:32 pm

.


Here is a different example of an irrational number which can be expressed as the infinite
sum of rational numbers:

sqrt(2) = 2^(1/2)


Using the Binomial Theorem, (a + b)^n = a^n + n*[a^(n - 1)]*b + [n(n - 1)/2]*[a^(n - 2)]*b^2 +

[n(n - 1)(n - 2)(n - 3)/6]*[a^(n - 3)]*b^3 + ...

Let a = b = 1.

And let n = 1/2.

Then the infinite series becomes 1 + 1/2 - 1/8 + 1/16 - ...

Starting from the second term, this is an alternating series.

The partial sum at any finite numbered term will never equal the square root of two,
because 1) the sum of a finite number of rational numbers is always rational and
2) the square root of two is irrational.

Sqrt(2) lies between (1 + 1/2) and (1 + 1/2 - 1/8), for example. Also, sqrt(2) lies
between (1 + 1/2 - 1/8) and (1 + 1/2 - 1/8 + 1/16).
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