When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia
The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. — Wikipedia
They're talking about infinite series, and saying that the limit is the sum of that series. I didn't quote the whole article, just the relevant part. Click the link and read for yourself. — Pfhorrest
The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the serie — Wikipedia
An infinite sum is defined because the mathematics community defined it; same as "twelve" is defined because English speakers defined it.An infinite sum is undefined because nobody has ever computed it. — EnPassant
...that's a term of art. It means to increase without bound. You're choking on mathematical language that you think represents some ideal thing that it just doesn't represent.tends to infinity — Wikipedia
...if I'm to understand correctly, the quibble you have is about what the "actual" infinite sum "actually" adds up to. Ironically, your quibble includes the notion that infinity is not a number. So I have no idea what you're talking about. Mathematicians define infinite sums differently.Narrator - What's the largest number you can think of?
Girl - uhm... hundred thousand?
Man - nine hundred and ninety nine thousand
Older man - A million
Narrator - In actual fact it's neither of these. The largest number is about forty five billion, although mathematicians suspect there are even larger numbers
Actual mathematicians do. — Pfhorrest
An infinite sum is defined because the mathematics community defined it; same as "twelve" is defined because English speakers defined it. — InPitzotl
That begs the question of what the heck you mean when you're talking about an infinite sum. Mathematicians regularly compute what they mean by it. It's the thing you're talking about that's nonsense, not the thing mathematicians are talking about.That still begs the question what is an infinite sum if nobody has ever computed it? — EnPassant
Case in point... what are you talking about?You can't jump from the finite to the infinite and expect finite rules to apply. — EnPassant
No, it's factual that it has been defined. Definitions aren't handed to us from an abstract guy giving out tablets in some Platonic/Pythagorean plane of existence. They're established by people... in this case, it's technical definitions given by mathematicians. They define it. You question that they define it, but that doesn't erase the fact that they, indeed, defined it.And it is questionable that it has been defined. — EnPassant
The infinite sum itself has been defined to be the limit... by mathematicians... who are the both the speakers of and designers of the language of math.All that has been rigorously defined is a limit. — EnPassant
They compute limits which are not the same as sums. A limit is what a finite sum converges to. — EnPassant
Given that the sum is by definition the limit, then by definition it's the same. You keep tripping over this same point.They compute limits which are not the same as sums — EnPassant
Apparently not... see the underlined as evidence for your continued confusion of the same point. The sum is by definition the same as the limit.I'm aware of that. — EnPassant
Right after the citation @Pfhorrest gave:But it has not been explicitly defined. — EnPassant
That is,
. — wikipedia
We can't know what an infinite sum is — EnPassant
You can't jump to infinity and expect the rules of finite arithmetic to apply — EnPassant
...and there's Meta's problem.
Family Resemblance. — Banno
This is a language barrier. In the language spoken by the mathematics community, .999... represents the same particular quantity that 1 does. — InPitzotl
Just for the heck of it, what are they, then? — tim wood
The diagonal of a square, for example, measured in the units that the sides are measured in, is how long? Is that length not a number? Or did something magic happen? — tim wood
If that's the case, then why have two distinct representations for one and the same thing? — Metaphysician Undercover
What do you mean "If that's the case, then"? There seems to be an implicit assumption that every thing should have exactly one name. Who exactly is making that assumption? It's not me, and it can't be you... does it say "Metaphysician Undercover" on your birth certificate? (And isn't 1 also equal to the fractions 1/1, 2/2, 3/3, and so on anyway?)If that's the case, then why have two distinct representations for one and the same thing? — Metaphysician Undercover
What matters to the present discussion is that .999... does not represent a number. Nor does .111... represent a number, and that's the problem with the op. — Metaphysician Undercover
As a matter of representing numbers, wouldn't most be fine with 9/9 = 9 × (1/9) = 9 × (0.111...) ? — jorndoe
It looks to go beyond this. Not only is 0.111... not a number, but there's no such thing as squares, because dimensions are incommensurable (@tim wood asked the question I was thinking before I got to it... and that was his response). Circles aren't real, so maybe trigonometry is a lie. Looks to me like Meta's a strange sort of Pythagorean?Is this Meta's claim? — Banno
Hmm, have you ever asked yourself a simple question "why"? Definitely we can, and we do, for centuries (since Leonard Euler's time) — Andrey Stolyarov
A true square does not admit to a diagonal, the two sides are incommensurable, making the square an irrational figure, just like the circle. There is no such thing as the diagonal of a square, because there is no such thing as a square, just like there's no such thing as a circle. These items were designed as ideals, but the irrationality of the ratios demonstrates that this effort was a failure. — Metaphysician Undercover
You can't jump to infinity and expect the rules of finite arithmetic to apply — EnPassant
Definitely we can, and we always do. Actually, this is all higher math (in contrast to elementary math) is all about. — Andrey Stolyarov
There's beautiful piece of math named "Functional analysis" (https://en.wikipedia.org/wiki/Functional_analysis), which works with spaces that have infinite number of dimensions — Andrey Stolyarov
It looks to go beyond this. — InPitzotl
I'd point out that 2 + 2 = 4, but we've previously determined that you don't even believe that. — fishfry
What do you mean "If that's the case, then"? There seems to be an implicit assumption that every thing should have exactly one name. — InPitzotl
So 1/9 is a number, even for Meta, but 0.111... is not? And this despite their being equal?
Is this Meta's claim? — Banno
And there you have it folks. MU is a genuine, triple-barreled whackdoodle. — tim wood
Did I say that I agree that 1/9 is a number? Check my definition of number, "particular quantity". How could 1/9 ever be construed as a particular quantity? A fraction is not a number. — Metaphysician Undercover
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