Is the convergence of an infinite series mere convention?

[pat8126]

It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality.

For instance, adding 1/2 + 1/4 + 1/8 . . . etc. would never become 1, since there would always be an infinitely small fraction that made the second half unreachable relative to the former fraction.

Unless, of course, there's a smallest value that eventually gets added twice at the end. Is that the implication in math or is it just rounded by convention regarding infinity?

Or is there some other reason?

 

[mfb]

It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality.

There is no such thing in the real numbers by definition. Mathematics is not about "reality", whatever that is supposed to mean, although mathematics is often useful in science.

 

[S.G. Janssens]

It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality.

Convergence is a precisely defined mathematical concept, independent of how reality works at the smallest scales.

 

[fresh_42]

I'm not sure I correctly understood you. Perhaps you would like to read about it:

http://platonicrealms.com/encyclopedia/zenos-paradox-of-the-tortoise-and-achilles
https://en.wikipedia.org/wiki/Zeno's_paradoxes#Archimedes

 

[pat8126]

Mathematics is not about "reality"

I suppose that was the largest misconception that I've had about mathematics.

 

[Mark44]

It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality.

For instance, adding 1/2 + 1/4 + 1/8 . . . etc. would never become 1

True as long as we're talking about a finite number of terms being added. But the sequence of partial sums, {1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, ...}, can be shown to get arbitrarily close to 1. This is the concept of convergence that @Krylov alluded to.

 

[pat8126]

True as long as we're talking about a finite number of terms being added

Eugene Wigner stated that "whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics." ("The Unreasonable Effectiveness of Mathematics in the Natural Sciences")

Could there be any inherent logical problems when mixing reality-based math with other types of math? For instance, the paradox created by Zeno deals with a finite concept that uses infinite concepts to solve it. It doesn't tackle the underlying question that math has been used to solve - the discrete nature of reality.

If one were to mix axioms that allow infinite definitions to the finite branches of mathematics, would there be any incompatibility between the question and the mathematical solution?

 

[fresh_42]

Eugene Wigner stated that "whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics." ("The Unreasonable Effectiveness of Mathematics in the Natural Sciences")

I disagree. Although Wigner probably would call a circle as "unquestionably true ... formulated to describe entities which are directly suggested by the actual world" nobody has ever managed to show me one.

Could there be any inherent logical problems when mixing reality-based math with other types of math? For instance, the paradox created by Zeno deals with a finite concept that uses infinite concepts to solve it. It doesn't tackle the underlying question that math has been used to solve - the discrete nature of reality.

Have you read this?

Mathematics is not about "reality", whatever that is supposed to mean, although mathematics is often useful in science.

If one were to mix axioms that allow infinite definitions to the finite branches of mathematics, would there be any incompatibility between the question and the mathematical solution?

I don't even know where to start to question these terms. But I suppose I'm not interested in the answers. At best, it is about philosophy, which we won't discuss here, since it always leads to nowhere. I recommend that you read what axioms are and find out yourself, what a "mix" could mean in this context.

To phrase a bottom line:

Infinity is nothing else, than the speechless answer of someone who proudly counted to ten and has been asked why he stopped.
The rest is history.

 

[pat8126]

I don't even know where to start to question these terms. But I suppose I'm not interested in the answers. At best, it is about philosophy, which we won't discuss here, since it always leads to nowhere. I recommend that you read what axioms are and find out yourself, what a "mix" could mean in this context."

I am not trying to create a philosophical discussion, as those are only allowed at the discretion of moderators. But I am trying to figure out if infinity is a mere convention, which it seems to be. In other words, it's just an axiomatic definition that governs by ipse dixit.

Please allow me to give a more concrete example of the mixing of axioms:

Let us define X as the sum of 1/2 + 1/4 + 1/8 . . . + 1/N as N approaches infinity.

Would X + 1 = 2 or would X + 1 < 2, for purposes of algebraic manipulation? Does the operator and equality sign transcend the set of numbers being compared and manipulated?

That's the mixing of math that I'm referencing. If you are offended by the question and truly do not care about the answer, please do not respond as it does not help me.

 

[fresh_42]

Let us define X as the sum of 1/2 + 1/4 + 1/8 . . . 1/N as 1, as N approaches infinity.

What does $\frac{1}{N}$ mean here?

Would X + 1 = 2 or would X + 1 < 2, for purposes of algebraic manipulation? Does the operator and equality sign transcend the set of numbers being compared and manipulated?

$$X_N +1 < 2 \text{ for }X_N:=\sum_{n=1}^{n=N}2^{-n} < 1 \text{ and any }N \in \mathbb{N} \text{ but }X_\infty :=\sum_{n=1}^{n=\infty}2^{-n}=\lim_{N\rightarrow \infty}\sum_{n=1}^{n=N}2^{-n} = 1$$

That's the mixing of math that I'm referencing. If you are offended by the question and truly do not care about the answer, please do not respond as it does not help me.

There is no mixing, only sloppy language.

 

[pat8126]

What does $\frac{1}{N}$ mean here?

Instead of 1/N, perhaps it should be 1/(2^N)

Zeno's paradox starts with the assumption that there are 2 finite points with a distance that can be halved, as one clearly can move from A to B. The question then dwells upon infinite halving, which seems to mix finite with infinite sets of numbers, thereby creating an impossible answer unless convergence comes into play, which is rounding, or there is a smallest possible distance.

As for language being sloppy, are you stating that my language is sloppy or language is sloppy in general? Or both? If the former, please check the forum rules as it involves insulting questioners.

 

[fresh_42]

As for language being sloppy, are you stating that my language is sloppy or language is sloppy in general? Or both? If the former, please check the forum rules as it involves insulting questioners.

Langauge in general is ambiguous. In addition it is difficult - for both of us - to only communicate by writings as est. 90% of communication is usually done by body language. I didn't want to offend you. I only wanted to demand precision as soon as the standard definitions and terminology are left, as "mixture of axioms" certainly is. I could explain axioms, but Wikipedia already did, and it is eventually controversial. And sorry, but a notation 1/2 + 1/4 + 1/8 . . . + 1/N is sloppy, because N suggests to be any natural number whereas the fractions only have powers of two in their denominator. Maybe I'm the stupid, but I couldn't clearly see what you meant by X either: the finite sum or the infinite sum? The latter being only a short form of the actual limit of a sequence of finite numbers. This really shouldn't offend you. So, sorry again, if it did. The sequence is completely smaller than one, although the limit is one, meaning the sequence comes as close to one, as close whatever is demanded.

The basic issues here is in my opinion the concept of infinity. It is artificial. However, it is natural at the same time which my counting example should demonstrated. And as mfb has said: mathematics isn't supposed to be "real" - whatever this means. It simply happens that it is a useful tool to describe / model / predict / invetsigate ... reality. In former times mathematics has actually counted as a Geisteswissenschaft (human science).

In order to handle infinity, mathematicians developed rigor concepts which help them to deal with it. You cannot simply transpose this to reality, in which it is hard to find something infinite. Therefore one has to be especially careful on the edge between the two.

 

[pat8126]

And sorry, but a notation 1/2 + 1/4 + 1/8 . . . + 1/N is sloppy

Thank you for the apology and I accept it. But in the future, know that you could simply have used the terms "imprecise" or "unclear" but chose to use an emotionally loaded adjective instead. That was the insulting part and it makes the entire point of the website moot, as no one wants to be afraid to ask questions. If people such as myself already knew the answers, clearly we wouldn't be seeking the answers from smarter and more educated people.

Concerning axioms, I believe they are defined to be truths that are so self-evident that they transcend the need for proof.

As for this thread, I suppose I should consider it closed as it answers my question. Convergence seems to be a rounding function by convention in the case of Zeno's paradox.

 

[Mark44]

As for language being sloppy, are you stating that my language is sloppy or language is sloppy in general? Or both? If the former, please check the forum rules as it involves insulting questioners.

I believe he was saying that the language you used was sloppy, specifically in how you defined the sum X. This is not a personal insult, so in no way violates forum rules.

As for this thread, I suppose I should consider it closed as it answers my question. Convergence seems to be a rounding function by convention in the case of Zeno's paradox.

Your question has been answered, but it doesn't seem to have altered your beliefs. Did you look at the link that Krylov supplied? Convergence has nothing to do with rounding. Zeno's Paradox (actually there are several paradoxes associated with Zeno) may have seemed paradoxical when it was first posed, but the flaw in that reasoning is that either space or time is made up of discrete intervals. Consider the arrow paradox, in which it is stated that an arrow can never reach its target. Anyone who has ever observed an arrow in flight can see that the reasonong behind this supposed paradox is flawed.

 

[pat8126]

I believe he was saying that the language you used was sloppy, specifically in how you defined the sum X. This is not a personal insult, so in no way violates forum rules.

Sloppy is defined, primarily, as " muddy, slushy, or very wet" - a very unflattering image appears in the mind's eye. Words are powerful and the particular choice made me cringe in regret that I posted. Perhaps I'm overly sensitive, but simply saying "imprecise" or "unclear" would seem more precise.

As for my question being answered, I'm afraid to delve further into any queries for fear of ridicule or having the post deleted for philosophy. But for what it's worth, how could summing that which is less than the previous half ever equal two halves unless it's a rounding at the boundaries of logic by the axiomatic conventions surrounding infinity?

I've taken third year calculus, linear algebra, discrete math, etc. and passed with A's in all of them back in my college years long ago, so please believe me that public ivy has granted me a wonderful education and I certainly understand an axiom.

I appreciate all the answers and realize that reality is either continuous and/or math does not represent reality. I considered it closed because I found interesting topics outside this thread by ideas posted within them.

 

[Mark44]

Sloppy is defined, primarily, as " muddy, slushy, or very wet" - a very unflattering image appears in the mind's eye. Words are powerful and the particular choice made me cringe in regret that I posted. Perhaps I'm overly sensitive, but simply saying "imprecise" or "unclear" would seem more precise.

There are other definitions for "sloppy" besides the ones you listed. In the context of this thread, "sloppy" is an apt description of what you wrote; namely

Let us define X as the sum of 1/2 + 1/4 + 1/8 . . . + 1/N as N approaches infinity.

A minor quibble is the last term being 1/N rather than following the pattern of the preceding terms. The sloppiness, IMO, is the part "as N approaches infinity." That does not define X adequately.

Perhaps you are being oversensitive. There's a difference between a criticism of what you say or write, as opposed to being a criticism of who you are. fresh_42's comment was about what you wrote. You should not take it as a personal criticism.

As for my question being answered, I'm afraid to delve further into any queries for fear of ridicule or having the post deleted for philosophy. But for what it's worth, how could summing that which is less than the previous half ever equal two halves unless it's a rounding by convention surrounding infinity?

I've taken third year calculus, linear algebra, discrete math, etc. and passed with A's in all of them back in my college years long ago, so please believe me that public ivy has granted me a wonderful education and I certainly understand an axiom.

We're not talking about axioms here -- the discussion is about what it means for the sum of a series with an infinite number of terms to converge. Again, I urge you to look at the link that Krylov gave earlier in this thread, in which the definition of convergence of a series is shown. You are mistaken about convergence being about rounding.

 

[pat8126]

We're not talking about axioms here -- the discussion is about what it means for the sum of a series with an infinite number of terms to converge. Again, I urge you to look at the link that Krylov gave earlier in this thread, in which the definition of convergence of a series is shown. You are mistaken about convergence being about rounding.

Yes, I did read that article and understand that convergence is a limit to a function. It's what an equation approaches, yet does not become, as it goes on forever. To me, that seems that such an approximation, as approaching does not mean equals, is a form of rounding. My usage of the verb "to round" is defined to mean "to alter (a number) to one less exact but more convenient for calculations." To say a series becomes that which it approaches for equality in later calculations is the logical axiom that I am referencing. It's a convention of thought.

As language is imprecise, perhaps I'm wrong in my usage and I apologize for the confusion. I am not able to properly express myself at the present time and once again, I am sorry if I am unable to comprehend what everyone is saying. I promise to think about the answers provided and hope to understand everyone's point of view in the future.

 

[Stephen Tashi]

Concerning axioms, I believe they are defined to be truths that are so self-evident that they transcend the need for proof.

That was a common viewpoint in the time of Euclid, but it is not the viewpoint used in contemporary mathematics.

To say that an axiom is a "truth" would imply that whatever it discussed (e.g. the number zero, an identity matrix, the limit of a sequence, etc.) has an existence apart from the formulation of the axiom. It is common for people, including some mathematicians, to have the "Platonic" view that mathematical objects do have such an existence. For example, people think "I know what zero is and the purpose of stating axioms about it is merely to list the things I know are true about it". However, even mathematicians who feel this way know that it is unreliable to use Platonic concepts in proofs. The modern concept is that axioms are assumptions. Different mathematical systems use different assumptions. If a particular real life situation is modeled by a mathematical system then we can consider whether it is true that the assumptions (i.e. axioms) of the particular system apply to the particular situation. That consideration is a matter of Science, not a matter of mathematics.

 

[Mark44]

Yes, I did read that article and understand that convergence is a limit to a function. It's what an equation approaches, yet does not become, as it goes on forever. To me, that seems that such an approximation, as approaching does not mean equals, is a form of rounding.

It's not at all rounding.

One way to think of convergence is as a dialog between two people -- one whose job it is to convince the other that a series converges to a certain number. Let's call them Prover and Skeptic.

Prover: "I can prove that $\frac 1 2 + \frac 1 4 + \frac 1 8 + \dots + \frac 1 {2^n} \dots$ converges to 1."
Skeptic: "I don't believe you. How can this be true?"
Prover: "I can make a finite sum of terms as close to 1 as you specify."
Skeptic: "Oh yeah? I'll bet you can't get it within 1/64 of 1."
Prover: "I'll use seven terms: $\frac 1 2 + \frac 1 4 + \frac 1 8 + \frac 1 {16} + \frac 1 {32} + \frac 1{64} + \frac 1 {128}$. These seven terms add up to 127/128, which is certainly within 1/64 of 1."
Skeptic: "Well, that's not really that close to 1. Can you get within .001 of 1?"
Prover: "Sure, this time I'll take ten terms in the sum. That us me to 1023/1024, which is clearly with .001 of 1."
Skeptic: "That's still not all that close. I'll bet you can't get it within .000001 (one-millionth)."
Prover: "I'll take 20 terms in the sum. That gets us to $\frac{1048575}{1048576}$, which is within one-millionth of 1."
Skeptic: "I guess no matter how close I say, you seem to be able to come up with a finite number of terms that have a sum that is closer. I give up -- you win."​

Notice that there is no rounding going on. The whole idea behind convergence is that no matter how close to the purported convergence limit someone else specifies, I can show that a finite number of terms in the sum comes closer to the limit. That's what it means to say that we can make the sum arbitrarily close to the limiting value.

My usage of the verb "to round" is defined to mean "to alter (a number) to one less exact but more convenient for calculations." To say a series becomes that which it approaches for equality in later calculations is the logical axiom that I am referencing. It's a convention of thought.

 

[Stephen Tashi]

Is this simply because infinite series are *defined* by their converged limit even though they only become infinitesimally close to it?

Yes, the sum of an infinite series is defined by the definition of a certain limit.

But No - There is nothing in that definition that specifies a process that proceeds step by step. There is nothing in the definition of that limit that defines a concept of "approaches" that takes place in time. The conception of an infinite series as something that is taking place in a series of steps and "approaching" some value is an intuitive idea that some people find useful, but it is not described within the definition of the sum of an infinite series. It is also not described within the definition of an infinite series itself.

Arguments based on the thought that an infinite series is a process that "becomes arbitrarily close" to some value are intuitive arguments. They aren't mathematical proofs. They are an example of applying mathematical "Platonism". People form a concept of an infinite series as something that is progressing in a series of steps and use it to guide their thinking about the properties of the infinite series that are specified by the definitions of mathematics. The controversies that arise when people reach different conclusions based on their Platonistic ideas of infinite series are a good example of why modern mathematics doesn't regard Platonistic notions as a basis for proofs.

 

[lurflurf]

Yes it is a convention.
I suggest we turn attention to the remainder
1-1/2- 1/4 - 1/8 . . .1/2^n=2^-n
In general we establish a convergence convention by deciding three things
1)what each thing converges to
2)what numbers we allow convergence to
3)when convergence occursThe "usual" convergence has be found to be convenient and useful in may situations but there are others

You have not described the problem you are solving for which usual convergence is not best
modify the above three choices
1)feel free to say
1/2 + 1/4 + 1/8 . . . =11
or
1/2 + 1/4 + 1/8 . . . =radish
or whatever if that fits your purpose
famously zero is the only zero like real number
so there is no other choice if we want a real number and usual addition
2)get some new numbers if you want they will not be real numbers and that may or may not be desired
3)fell free to say
1/2 + 1/4 + 1/8 . . . diverges
if that suits your purpose

The disconnect is that when someone says
1/2 + 1/4 + 1/8 . . . =1
they mean it in the context of particular rules
there is nothing to question this is well established
if someone says
1/2 + 1/4 + 1/8 . . . !=1
they either do not understand the rules or they are using different rules
they should explain their position

As far as if it should be called rounding or not I think not, but it is an unimportant side issue.
Sometimes we think of a sequence as a real number then it is not rounding.
Sometimes we think of a sequence as something other than a real number then it is rounding.
It does not really matter.

 

[lurflurf]

It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality.

What?

There is nothing in that definition that specifies a process that proceeds step by step.

There are many definition of limit and some imply or out right state a step by step process.

Where is this from?
"Is this simply because infinite series are *defined* by their converged limit even though they only become infinitesimally close to it?"
The value of a limit is it uses imprecise descriptions to describe precisely.

A (convergent) limit says there is this one number and here are an infinite number of estimates. The collection of estimated determines the number while individual estimates may not. The estimates define the number but they are not the number.

If Robertha is sqrt(3) meters tall the fact I say
1,1.7,1.73,1.732,1.732,1.73205,...->Robertha's height/meters
Does not mean we her height cannot be reached.

 

[Stephen Tashi]

There are many definition of limit and some imply or out right state a step by step process.

I'm referring to the standard definition for the sum of a series.

 

[pat8126]

Yes, the sum of an infinite series is defined by the definition of a certain limit.

But No - There is nothing in that definition that specifies a process that proceeds step by step. There is nothing in the definition of that limit that defines a concept of "approaches" that takes place in time.

I think that I finally understand the source of my confusion, thanks to everyone.

I have been equating the sum of the series with the value to which the series converges. The process to find the limit requires a step-by-step process, but the final value for further use in equations is the limit value itself.

Therefore, it makes sense to say the limit of my example is 1, while the sum of the series is not quite 1. As such, any algebraic manipulation with the sum of the series makes perfect sense because the relevant value of the series' summation is the limit.

 

[pwsnafu]

Therefore, it makes sense to say the limit of my example is 1, while the sum of the series is not quite 1.

The finite partial sums are not quite 1.

As such, any algebraic manipulation with the sum of the series makes perfect sense because the relevant value of the series' summation is the limit.

Some manipulations are valid. See the rearrangement theorem for a counter-example.

 

[Mark44]

I have been equating the sum of the series with the value to which the series converges.

The sum of a (convergent) series is the value to which the series converges. I think that your confusion might be around not distinguishing between a partial sum (with a finite number of terms) and the series itself, which contains an infinite number of terms.

The process to find the limit requires a step-by-step process, but the final value for further use in equations is the limit value itself.

Therefore, it makes sense to say the limit of my example is 1, while the sum of the series is not quite 1.

No, this doesn't make sense. $\lim_{n \to \infty}\frac 1 2 + \frac 1 4 + \frac 1 8 + \dots + \frac 1 {2^n} = 1$, however, the partial sum $\frac 1 2 + \frac 1 4 + \frac 1 8 + \dots + \frac 1 {2^n}$ is always a little less than 1. This latter sum is a partial sum, and has a finite number of terms.

A series converges if its sequence of partial sums converges. For the series in this thread, the sequence of partial sums is $\{\frac 1 2, \frac 3 4, \frac 7 8, \dots, \frac{2^n - 1} {2^n}\}$. If the sequence of partial sums converges, the series itself converges to the same number.

As such, any algebraic manipulation with the sum of the series makes perfect sense because the relevant value of the series' summation is the limit.

 

[pat8126]

I think that your confusion might be around not distinguishing between a partial sum (with a finite number of terms) and the series itself, which contains an infinite number of terms.

Yes, that makes sense. I was not properly understanding the terminology differences between partial sums and series.

The partial sum is the addition of every part of the series from the lowest bound to some particular finite upper bound. The limit to which the series converges is based upon the pattern found in the sequence of partial sums as it continues infinitely. Is that correct?

 

[Mark44]

Yes, that makes sense. I was not properly understanding the terminology differences between partial sums and series.

The partial sum is the addition of every part of the series from the lowest bound to some particular finite upper bound. The limit to which the series converges is based upon the pattern found in the sequence of partial sums as it continues infinitely. Is that correct?

That's not too far off. I wouldn't say "the pattern found in the sequence of partial sums" -- I would just say "the limit, as n grows large, of the sequence of partial sums."

 

[pat8126]

I would just say "the limit, as n grows large, of the sequence of partial sums."

Thank you, I see what you mean. I was attempting to use the word "pattern" to describe ways to calculate the limit, such as a series that telescoped. But you are of course correct - "the limit, as n grows large, of the sequence of partial sums" is much more precise.

 

[votingmachine]

I don't know if it helps, but consider the sum:
0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 ...
This adds up to the repeating decimal: 0.33333...

Most people can run thru the proof of why this is EQUAL to one-third (if you are unfamiliar just ask). Even though it is an infinitely long sum, and at what ever finite level you look, there is always another "3".

 

[jbriggs444]

Even though it is an infinitely long sum, and at what ever finite level you look, there is always another "3".

It is a convention to call this a "sum", and in many ways it acts like one, but it is not an infinitely long sum. Instead it is the limit of the sequence of partial sums.

 

[pat8126]

Where is this from?
"Is this simply because infinite series are *defined* by their converged limit even though they only become infinitesimally close to it?"

That was from a previous posting that I deleted. It was starting to explore the philosophy of mathematics so I removed it accordingly.