Just like 1+1 is another way of writing the number 2, 0.333… is another way of writing the number 1/3.
Its not quite the same. You can actually write 1+1. But you cannot write infinitely many 3's (without using some other form of notation, at least)
And BTW, I do know what I'm talking about. As a junior in high school, I was written a letter of rec by my math teacher last year to be one of the very few students to skip a year of math, and as one of the 2 juniors in calc bc (at a school of over 3000 students), I am toward the top of my class.
Its not quite the same. You can actually write 1+1. But you cannot write infinitely many 3's (without using some other form of notation, at least)
And BTW, I do know what I'm talking about. As a junior in high school, I was written a letter of rec by my math teacher last year to be one of the very few students to skip a year of math, and as one of the 2 juniors in calc bc (at a school of over 3000 students), I am toward the top of my class.
o.m.g. wow. I’m so humbled to be posting in the same forum as someone so pedigreed.
No, that’s just silly and wrong when that series is interpreted in its most natural meaning. The article tells you why. Ramanujan summation has other applications, but it doesn’t apply to compute this sum that is known to diverge.
It’s not that hard to believe if it’s redefined as something it is not.
It is not "redefined as something it is not". It has been mathematically proven multiple ways, and has been a known fact for hundreds of years. As I previously stated before, this value and proof is not only in textbooks, but also has applications in real life physics. It is not some figment of my imagination. I agree, it seems completely stupid, if you start with a positive value and only add positive values how can it be negative? But it is. I'll show you one such proof. If you claim this is incorrect, find my mistake. I will label it step by step.
1) (S1) = 1-1+1-1+1-1+1-1+1-1+1... = 1/2 I don't want to have to prove this series as well, feel free to look it up.
2) (S2) = 1-2+3-4+5-6+7...
3) (S3) = 1+2+3+4+5+6+7...
4) Lets add S2+S2. Because it is all addition and subtraction, the communicative property states it does not matter the order we add them. All I did was start adding them shifted on number over, which is perfectly fine.
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8...
+ 1 - 2 + 3 - 4 + 5 - 6 + 7... (The next line is the sum of the two series)
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ... and we know this equals 1/2
5) given that S2 + S2 = 1/2, we can divide both sides by 2 to find that S2 = 1/4
6) Lets take S3, and subtract S2. If we subtract the entire sequence, we can simply switch all the signs, so subtracting -2 is the same as adding 2. I will automatically do this to not waste space.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 ... (again note that I have switched all the signs in the next line)
- 1 + 2 - 3 + 4 - 5 + 6 - 7 + 8 ... (The next line is the sum of the two series)
0+ 4 + 0 + 8 + 0 +12 + 0+ 16...
7) If you keep calculating it out, you will find that S3 - S2 = 4 + 8 + 12 + 16 + 20...
8) All of these are multiples of 4, so you can factor out the 4. S3 - S2 = 4 (1 + 2 + 3 + 4 + 5...)
9) That series is equivalent to S3! So now you have a simple equation. S3 - S2 = 4(S3).
10) S2 = 1/4. So we know S3 - 1/4 = 4(S3)
11) Subtract S3 from both sides. -1/4 = 3(S3)
12) Divide both sides by 3. -1/12 = S3
And there you have it! One of the simpler proofs to understand this (though there are many others).
1+2+3+4+5+6+7... = - 1/12
Find the mistake, and if you want, I can give you another proof of this.
You are a kid in school. I have no interest in pointing out your stupidity, sorry. You are wrong in your claims and don’t seem to even realize why.
Its not quite the same. You can actually write 1+1. But you cannot write infinitely many 3's (without using some other form of notation, at least)
And BTW, I do know what I'm talking about. As a junior in high school, I was written a letter of rec by my math teacher last year to be one of the very few students to skip a year of math, and as one of the 2 juniors in calc bc (at a school of over 3000 students), I am toward the top of my class.
Every mathematician agrees that 0.333.... equals 1/3. They represent the same point on a number line. Linguistics have nothing to do with it. These are two different representations of the same rational number.
I upvoted you because that is exactly my point. The fact that what we "say" in "normal life" has no bearing on the math is what I am commenting on.
That the math can be "true" without it changing our intuitive understanding of the world or how we talk about the world.
The physical reality of the Earth's rotation has no impact on how we talk about sunrise and sunset, for example. The sense of a flat Earth, for example, feels true but isn't.
The way humans perceive the world is not always the same as how the world really is.
This post was edited 1 minute after it was posted.
No dude. You’re just being unnecessarily pedantic and obtuse over something you don’t understand. The curse of the know-it-all, if you will. You are wrong but your enormous ego doesn’t allow you accept that there’s just something you’re not understanding here.
There is no “infinitely close” here. You’ve invented that whole cloth in an attempt to take the other side because you love the smell of your own farts just that much.
Just like 1+1 is another way of writing the number 2, 0.333… is another way of writing the number 1/3.
My god are you insufferable.
You are probably right on this one... I wasn't trying to be a know-it-all (since I don't). Sorry about that. I am not 100% sure why everyone is enraged at me on this one, but I blame myself since you guys can't all be crazy. :)
And "infinitely close" was my shorthand for the definition of an asymptote, the term I use for approaching a value but never reaching it. Was I wrong to use this definition? "An asymptote is a value that gets infinitely close to, but never actually touches or crosses number on a graph."
I do think the best post was from the "Real Theoretical Math Guy" who said that the sum of an infinite series will never equal a single, actual whole number such as 1 since the summing never ends.
That is what I meant (in my very first post) about the word "equal" having two meanings. In some languages they have different words for equal (in value) and equal (the same). In German, for example, four Quarters is das Gleiche wie one Dollar but it is not das Selbe wie one Dollar.
This post was edited 7 minutes after it was posted.
It is not "redefined as something it is not". It has been mathematically proven multiple ways, and has been a known fact for hundreds of years. As I previously stated before, this value and proof is not only in textbooks, but also has applications in real life physics. It is not some figment of my imagination. I agree, it seems completely stupid, if you start with a positive value and only add positive values how can it be negative? But it is. I'll show you one such proof. If you claim this is incorrect, find my mistake. I will label it step by step.
1) (S1) = 1-1+1-1+1-1+1-1+1-1+1... = 1/2 I don't want to have to prove this series as well, feel free to look it up.
2) (S2) = 1-2+3-4+5-6+7...
3) (S3) = 1+2+3+4+5+6+7...
4) Lets add S2+S2. Because it is all addition and subtraction, the communicative property states it does not matter the order we add them. All I did was start adding them shifted on number over, which is perfectly fine.
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8...
+ 1 - 2 + 3 - 4 + 5 - 6 + 7... (The next line is the sum of the two series)
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ... and we know this equals 1/2
5) given that S2 + S2 = 1/2, we can divide both sides by 2 to find that S2 = 1/4
6) Lets take S3, and subtract S2. If we subtract the entire sequence, we can simply switch all the signs, so subtracting -2 is the same as adding 2. I will automatically do this to not waste space.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 ... (again note that I have switched all the signs in the next line)
- 1 + 2 - 3 + 4 - 5 + 6 - 7 + 8 ... (The next line is the sum of the two series)
0+ 4 + 0 + 8 + 0 +12 + 0+ 16...
7) If you keep calculating it out, you will find that S3 - S2 = 4 + 8 + 12 + 16 + 20...
8) All of these are multiples of 4, so you can factor out the 4. S3 - S2 = 4 (1 + 2 + 3 + 4 + 5...)
9) That series is equivalent to S3! So now you have a simple equation. S3 - S2 = 4(S3).
10) S2 = 1/4. So we know S3 - 1/4 = 4(S3)
11) Subtract S3 from both sides. -1/4 = 3(S3)
12) Divide both sides by 3. -1/12 = S3
And there you have it! One of the simpler proofs to understand this (though there are many others).
1+2+3+4+5+6+7... = - 1/12
Find the mistake, and if you want, I can give you another proof of this.
You are a kid in school. I have no interest in pointing out your stupidity, sorry. You are wrong in your claims and don’t seem to even realize why.
A kid in school who likely knows a whole lot more math than you.
And again, if I am wrong in my claims, where? You can't seem to point out a mistake, or provide a mathematical proof to demonstrate I am wrong.
You are a kid in school. I have no interest in pointing out your stupidity, sorry. You are wrong in your claims and don’t seem to even realize why.
A kid in school who likely knows a whole lot more math than you.
And again, if I am wrong in my claims, where? You can't seem to point out a mistake, or provide a mathematical proof to demonstrate I am wrong.
Also, in other threads I participate in (as well as irl), people can't stop bringing up "textbooks say xy is boy so you must be a boy (but lets not get into this). So why is it that now you are falsely claiming that this fact is false when it has been used in many physics textbooks for many years? Why is it that you are falsely claiming that some of the best mathematicians who have ever lived are incorrect about this? There are many, MANY, mathematical proofs proving that the sum of all natural numbers equals -1/12, and this fact has been known for centuries. To me, it seems that people can't possibly imagine themselves being wrong, so they will never listen to cold, hard, facts.
So again, if you think you are so d*mn smart and I am just a dumb kid, disprove my claim. You can't.
They are exactly equal. Any finite number of digits of 0.333333.... can get you arbitrarily close to 1/3, but that's not the question. If you don't limit yourself to a finite sequence (and you don't have to) then it's 1/3.
If it's not 1/3 then there must be a number between 0.33333... and 1/3 (because that's how real numbers work. Between any two distinct real numbers there is a real number distinct from both of them). Give me such a number.
the argument on this thread is between those who understand ".3 + .03 + .003 + .0003 + ..." refers to the limit of that expression, and those who think the expression itself represents a quantity. So it's pointless, like the memes playing on the division sign to confuse order of operations.
It's an airplane on a treadmill
That distinction between 'limit of that expression' and 'a quantity' is not real.
A limit, where it exists (as it does here, since the series converges), is just a number, by definition. You can simplify it if necessary, but even if you don't it is still a sequence of symbols that represents a number. When you write (lim n->infinity f(n)), if the limit exists, that is just an expression representing a number the way 12 or (1+23*-7) or sin(301.5) or e^29.1 each represent numbers.
If 0.333... weren't a number, the equation 0.333... = 1/3 would be neither true nor false, it would be something akin to an apples-to-oranges comparison. But limits aren't some weird non-number object that is not comparable to other numbers; the whole point of them is to be numbers.
This is one of the best responses. The repeating values in 0.33333... are just an artifact of our choice of base-10 (decimal) system to express the number. And 0.333... in base 10 is provably and exactly equal to 1/3 as long as there are truly an infinite amount of 3's.
No, that’s just silly and wrong when that series is interpreted in its most natural meaning. The article tells you why. Ramanujan summation has other applications, but it doesn’t apply to compute this sum that is known to diverge.
It’s not that hard to believe if it’s redefined as something it is not.
It is not "redefined as something it is not". It has been mathematically proven multiple ways, and has been a known fact for hundreds of years. As I previously stated before, this value and proof is not only in textbooks, but also has applications in real life physics. It is not some figment of my imagination. I agree, it seems completely stupid, if you start with a positive value and only add positive values how can it be negative? But it is. I'll show you one such proof. If you claim this is incorrect, find my mistake. I will label it step by step.
1) (S1) = 1-1+1-1+1-1+1-1+1-1+1... = 1/2 I don't want to have to prove this series as well, feel free to look it up.
2) (S2) = 1-2+3-4+5-6+7...
3) (S3) = 1+2+3+4+5+6+7...
4) Lets add S2+S2. Because it is all addition and subtraction, the communicative property states it does not matter the order we add them. All I did was start adding them shifted on number over, which is perfectly fine.
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8...
+ 1 - 2 + 3 - 4 + 5 - 6 + 7... (The next line is the sum of the two series)
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ... and we know this equals 1/2
5) given that S2 + S2 = 1/2, we can divide both sides by 2 to find that S2 = 1/4
6) Lets take S3, and subtract S2. If we subtract the entire sequence, we can simply switch all the signs, so subtracting -2 is the same as adding 2. I will automatically do this to not waste space.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 ... (again note that I have switched all the signs in the next line)
- 1 + 2 - 3 + 4 - 5 + 6 - 7 + 8 ... (The next line is the sum of the two series)
0+ 4 + 0 + 8 + 0 +12 + 0+ 16...
7) If you keep calculating it out, you will find that S3 - S2 = 4 + 8 + 12 + 16 + 20...
8) All of these are multiples of 4, so you can factor out the 4. S3 - S2 = 4 (1 + 2 + 3 + 4 + 5...)
9) That series is equivalent to S3! So now you have a simple equation. S3 - S2 = 4(S3).
10) S2 = 1/4. So we know S3 - 1/4 = 4(S3)
11) Subtract S3 from both sides. -1/4 = 3(S3)
12) Divide both sides by 3. -1/12 = S3
And there you have it! One of the simpler proofs to understand this (though there are many others).
1+2+3+4+5+6+7... = - 1/12
Find the mistake, and if you want, I can give you another proof of this.
The mistake is in (1). S1 is not equal to 1/2. It is a divergent series that does not have a finite limit. If you haven't covered epsilon-delta (or epsilon-N) proofs, learn the appropriate definitions of limits and why they do not apply to this case.
(yes, yes, I know this is bait and that's why you said "communicative" instead of "commutative" in (4))
It is not "redefined as something it is not". It has been mathematically proven multiple ways, and has been a known fact for hundreds of years. As I previously stated before, this value and proof is not only in textbooks, but also has applications in real life physics. It is not some figment of my imagination. I agree, it seems completely stupid, if you start with a positive value and only add positive values how can it be negative? But it is. I'll show you one such proof. If you claim this is incorrect, find my mistake. I will label it step by step.
1) (S1) = 1-1+1-1+1-1+1-1+1-1+1... = 1/2 I don't want to have to prove this series as well, feel free to look it up.
2) (S2) = 1-2+3-4+5-6+7...
3) (S3) = 1+2+3+4+5+6+7...
4) Lets add S2+S2. Because it is all addition and subtraction, the communicative property states it does not matter the order we add them. All I did was start adding them shifted on number over, which is perfectly fine.
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8...
+ 1 - 2 + 3 - 4 + 5 - 6 + 7... (The next line is the sum of the two series)
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ... and we know this equals 1/2
5) given that S2 + S2 = 1/2, we can divide both sides by 2 to find that S2 = 1/4
6) Lets take S3, and subtract S2. If we subtract the entire sequence, we can simply switch all the signs, so subtracting -2 is the same as adding 2. I will automatically do this to not waste space.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 ... (again note that I have switched all the signs in the next line)
- 1 + 2 - 3 + 4 - 5 + 6 - 7 + 8 ... (The next line is the sum of the two series)
0+ 4 + 0 + 8 + 0 +12 + 0+ 16...
7) If you keep calculating it out, you will find that S3 - S2 = 4 + 8 + 12 + 16 + 20...
8) All of these are multiples of 4, so you can factor out the 4. S3 - S2 = 4 (1 + 2 + 3 + 4 + 5...)
9) That series is equivalent to S3! So now you have a simple equation. S3 - S2 = 4(S3).
10) S2 = 1/4. So we know S3 - 1/4 = 4(S3)
11) Subtract S3 from both sides. -1/4 = 3(S3)
12) Divide both sides by 3. -1/12 = S3
And there you have it! One of the simpler proofs to understand this (though there are many others).
1+2+3+4+5+6+7... = - 1/12
Find the mistake, and if you want, I can give you another proof of this.
The mistake is in (1). S1 is not equal to 1/2. It is a divergent series that does not have a finite limit. If you haven't covered epsilon-delta (or epsilon-N) proofs, learn the appropriate definitions of limits and why they do not apply to this case.
(yes, yes, I know this is bait and that's why you said "communicative" instead of "commutative" in (4))
When using calculus, the limit does not converge. But S1 does equal 1/2. I apologize for the typo, this is not bait. Here is one of many proofs that the infinite series of 1-1+1-1+1-1+1-1+1...=1/2
1) Lets say S = 1-1+1-1+1-1+1-1+1...
2) Take 1-S
3) 1 - (1-1+1-1+1-1+1-1+1...) = 1 + (-1+1-1+1-1+1-1+1-1...) (Because we are subtracting every term in the brackets, I just flipped the signs.
4) So now we have 1-1+1-1+1-1+1-1+1-1...
5) But that is the same as S
6) So now we have an equation. 1 - S = S
7) Add S to both sides. 1 = 2S
8) So S=1/2
Again, this is a relatively well known proof. Also, yes, I understand the definitions of limits. And, using these definitions, I agree that there is no solution, just as I agree that using classical limit rules the sum of all natural numbers approaches infinity. However, there are multiple ways to solve math problems, and some result in different answers, but that does not invalidate them. It is similar to how the quantum mechanics and the theory of gravity have not been united. Does that mean one study is based on lies? Of course not. In this problem, using calculus we find that that there is no solution, but by using a bit of number theory we find a new, completely valid, solution.
You said it yourself, that the limit doesn't exist using calculus, but can exist using other methods. The problem is that when there are multiple ways to assign a value to an infinite series, you need to formally define what "=" means.
Simply put, the truth is that we have here is 2 different functions that both take in an infinite series and assigns a value to them. One is taking the limit in the sequence of partial sums, and the other is using the zeta function regularization to assign values to sums like 1 - 1 + 1 - 1 + ..., 1 - 2 + 3 - 4 + ..., and 1 + 2 + 3 + 4 + ..., etc. There are also other methods like cesaro summation, which can assign a value to 1 - 1 + 1 - 1 + ... but fails for series like sum_n=1 ^inf (n).
The first method would fail to converge for all of the series I mentioned above, but the second method does assign a value to them, and is equal to the first method whenever the partial sums do converge. However, this says nothing about the "value" of the summation, because if you want to talk about the sum of an infinite series, you need to formally define what it means for an infinite sum to exist.
In particular, what is mathematically proven to be incorrect is expecting the sum to not change when manipulating the terms of an infinite series that isn't absolutely convergent. In my previous comment, I was able to arrive at a different sum, and in fact, even for convergent series like 1 - 1/2 + 1/3 - 1/4 + ..., I can rearrange the terms to have the sum converge to any real value. This is coming from someone who knows a good amount of complex and real analysis.
This post was edited 2 minutes after it was posted.
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