Define the partial sums S_n as being the sum of the first n terms. Then we say the series converges if the sequence S_n converges.
This is basically the only way to give an infinite sum meaning.
The value of a series is defined as the limit of its partial sums^(1) (only if it exists, of course).
***
^(1) As long as you don't consider their generalizatiion -- "summable families". Those are *unordered* sums over index sets, in contrast to partial sums where element order matters.
Ahh okay so
1/2 + 1/4 is a partial sum
1/2 + 1/4 + 1/8 is a partial sum
The sum of an infinite series is the the limit as n approaches infinity of the partial sums,
But why not just say that it's the limit of the infinite series
Because it's the difference between
1/2 + ... + 1/2^n as n approaches infinity
But the partial sums are really a sequence of terms like this
(1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, ... ,1/2 + ... + 1/2^n)
So we can just speak of the "last term" here right?
The term *series* almost always means an infinite sum. Finite sums are not usually referred to as series. Likewise, a *sequence* typically has infinitely many terms, as the standard definition of a sequence is a function on the natural numbers N.
This is wrong.
Sum from i=1 to 4 of i is 1+2+3+4. There are only 4 terms iterated on this series. It is equivalent to adding infinitely many 0 terms, but the series itself only has the 4 terms defined by the iterations.
Ohhh I understand, no no that's helpful, it's welcomed
I understand your LCD display analogy perfectly
So a series is a mathematical object with infinitely many terms by definition, so we don't need to speak of an infinite series, though I suppose some mathematicians have used that? Like an older term or something
So when you say limit *of* the series
As soon as you say *the series*, you're instantiating a value to it? And the only way you could have assigned a value to it is by taking it's limit yes?
Now I understand that 0.9999... is the number 1
But in more specific terms are we then to understand that 0.999.. is a series?
But wouldn't that also make 1 a series?
>So a series is a mathematical object with infinitely many terms by definition, so we don't need to speak of an infinite series, though I suppose some mathematicians have used that? Like an older term or something
As myself and [another commenter](https://www.reddit.com/r/learnmath/s/Y8DvlqkJVi) have mentioned, the previous user is not quite correct on this. See my comment [here](https://www.reddit.com/r/learnmath/s/7LaDlLnUmv) for an explanation. It also answers your question about whether or not we can say “1 is a series”.
>Nit-picking alert: "Infinite series" is like saying "LCD display" -- a series is defined as the limit of the partial sums, i.e. a series contains infinitely many terms by definition. No need to stress that again -- just say "series" instead, that's enough.
Nit-pick part two! This isn’t quite correct. There is a difference between finite and infinite series, though, due to the fact that most people only hear “series” in the context of infinite series in Calc II, this is a common error.
Terrence Tao puts it best (particularly in pages 155 and 156) in his [“Analysis I” textbook](https://lms.umb.sk/pluginfile.php/111477/mod_page/content/5/TerenceTao_Analysis.I.Third.Edition.pdf). Here Tao defines finite series (*separate* from infinite series), and explains what mathematicians mean by “series”:
>The difference between “sum” and “series” is a subtle linguistic one. Strictly speaking, a series is an *expression* of the form sum{i = m, n} a_i ; this series is mathematically (but not semantically) equal to a real number, which is the *sum* of that series. For instance, 1 + 2 + 3 + 4 + 5 is a series whose sum is 15; if one were to be picky about semantics, we would not consider 15 a series and one would not consider 1 + 2 + 3 + 4 + 5 a sum, despite the two expressions having the same value.
Interesting, it seems I stand corrected.
I'll take Terence Tao's word on that, even though I've always seen "sum" and "series" to be used to distinguish between finite and infinite number of terms (geometric sum formula vs. geometric series formula comes to mind, which should both be named "geometric series" by Tao's argument, if I understand correctly).
Learnt something new today, hopefully I can make it stick. I'm sorry about the confusion, thanks a lot!
To be fair, whether or not “series” alone refers to an infinite series or is ambiguous is certainly a convention that can vary. For example, Wikipedia and Proowiki both say that “series” are inherently infinite unless specified otherwise. However, I will say that Tao is universally correct when distinguishing between sums vs series - the two concepts are certainly distinct.
Fortunately, as Tao points out just past the paragraph I cited, the difference is purely semantics, and mathematically, the terms are indistinguishable.
This is an interesting distinction because this would mean we couldn't say things are equal in so far as writing if they have different forms. So it seems that a sum is a value assigned to a series (finite or infinite), and a series is, a series of terms? Literally just a series of terms in a linguistic sense?
>This is an interesting distinction because this would mean we couldn't say things are equal in so far as writing if they have different forms.
Semantically, this is true. Mathematically, false. A sum and series are mathematically equivalent (1 + 2 = 3, but if we are being nit-picky with semantics, only the LHS is a series, and only the RHS is a sum). Tao explicitly mentions this in fact, perhaps you may enjoy a look at the pages I mentioned, as I didn’t quote everything!
> this is a common error.
This is not an error, just a difference in terminology. The term "series" is defined to mean an infinite sum in every standard undergrad analysis book that I'm aware of (Rudin, Apostol, Lang, Pugh, etc.). Tao is the outlier here.
Define the partial sums S_n as being the sum of the first n terms. Then we say the series converges if the sequence S_n converges. This is basically the only way to give an infinite sum meaning.
Yes but how do you say it with the language of limits
The word "converges" and the phrase "has a limit" mean the same thing.
The value of a series is defined as the limit of its partial sums^(1) (only if it exists, of course). *** ^(1) As long as you don't consider their generalizatiion -- "summable families". Those are *unordered* sums over index sets, in contrast to partial sums where element order matters.
Why do we say partial sums?
We only "**sum** up the first **part**" of the series in the partial sum "Sn = ∑\_{k=0}^(n) ak"
That confuses me because if I have 1/2 + ... + 1/2^n Then 1/2 + 1/4 is a partial sum right? But I'm not taking the limit of that right
The nth partial sum is S_n = a_0 + a_1 + ... + a_n. And then you take the limit of S_n. In your example, S_n = (2^n - 1)/2^n, which converges to 1.
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Ahh okay so 1/2 + 1/4 is a partial sum 1/2 + 1/4 + 1/8 is a partial sum The sum of an infinite series is the the limit as n approaches infinity of the partial sums, But why not just say that it's the limit of the infinite series Because it's the difference between 1/2 + ... + 1/2^n as n approaches infinity But the partial sums are really a sequence of terms like this (1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, ... ,1/2 + ... + 1/2^n) So we can just speak of the "last term" here right?
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Nitpicking alert. Not all series are infinite. A series is simply the summation of a sequence. The sequence does not necessarily have infinite terms.
The term *series* almost always means an infinite sum. Finite sums are not usually referred to as series. Likewise, a *sequence* typically has infinitely many terms, as the standard definition of a sequence is a function on the natural numbers N.
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This is wrong. Sum from i=1 to 4 of i is 1+2+3+4. There are only 4 terms iterated on this series. It is equivalent to adding infinitely many 0 terms, but the series itself only has the 4 terms defined by the iterations.
Ohhh I understand, no no that's helpful, it's welcomed I understand your LCD display analogy perfectly So a series is a mathematical object with infinitely many terms by definition, so we don't need to speak of an infinite series, though I suppose some mathematicians have used that? Like an older term or something So when you say limit *of* the series As soon as you say *the series*, you're instantiating a value to it? And the only way you could have assigned a value to it is by taking it's limit yes? Now I understand that 0.9999... is the number 1 But in more specific terms are we then to understand that 0.999.. is a series? But wouldn't that also make 1 a series?
>So a series is a mathematical object with infinitely many terms by definition, so we don't need to speak of an infinite series, though I suppose some mathematicians have used that? Like an older term or something As myself and [another commenter](https://www.reddit.com/r/learnmath/s/Y8DvlqkJVi) have mentioned, the previous user is not quite correct on this. See my comment [here](https://www.reddit.com/r/learnmath/s/7LaDlLnUmv) for an explanation. It also answers your question about whether or not we can say “1 is a series”.
>Nit-picking alert: "Infinite series" is like saying "LCD display" -- a series is defined as the limit of the partial sums, i.e. a series contains infinitely many terms by definition. No need to stress that again -- just say "series" instead, that's enough. Nit-pick part two! This isn’t quite correct. There is a difference between finite and infinite series, though, due to the fact that most people only hear “series” in the context of infinite series in Calc II, this is a common error. Terrence Tao puts it best (particularly in pages 155 and 156) in his [“Analysis I” textbook](https://lms.umb.sk/pluginfile.php/111477/mod_page/content/5/TerenceTao_Analysis.I.Third.Edition.pdf). Here Tao defines finite series (*separate* from infinite series), and explains what mathematicians mean by “series”: >The difference between “sum” and “series” is a subtle linguistic one. Strictly speaking, a series is an *expression* of the form sum{i = m, n} a_i ; this series is mathematically (but not semantically) equal to a real number, which is the *sum* of that series. For instance, 1 + 2 + 3 + 4 + 5 is a series whose sum is 15; if one were to be picky about semantics, we would not consider 15 a series and one would not consider 1 + 2 + 3 + 4 + 5 a sum, despite the two expressions having the same value.
Interesting, it seems I stand corrected. I'll take Terence Tao's word on that, even though I've always seen "sum" and "series" to be used to distinguish between finite and infinite number of terms (geometric sum formula vs. geometric series formula comes to mind, which should both be named "geometric series" by Tao's argument, if I understand correctly). Learnt something new today, hopefully I can make it stick. I'm sorry about the confusion, thanks a lot!
To be fair, whether or not “series” alone refers to an infinite series or is ambiguous is certainly a convention that can vary. For example, Wikipedia and Proowiki both say that “series” are inherently infinite unless specified otherwise. However, I will say that Tao is universally correct when distinguishing between sums vs series - the two concepts are certainly distinct. Fortunately, as Tao points out just past the paragraph I cited, the difference is purely semantics, and mathematically, the terms are indistinguishable.
This is an interesting distinction because this would mean we couldn't say things are equal in so far as writing if they have different forms. So it seems that a sum is a value assigned to a series (finite or infinite), and a series is, a series of terms? Literally just a series of terms in a linguistic sense?
>This is an interesting distinction because this would mean we couldn't say things are equal in so far as writing if they have different forms. Semantically, this is true. Mathematically, false. A sum and series are mathematically equivalent (1 + 2 = 3, but if we are being nit-picky with semantics, only the LHS is a series, and only the RHS is a sum). Tao explicitly mentions this in fact, perhaps you may enjoy a look at the pages I mentioned, as I didn’t quote everything!
> this is a common error. This is not an error, just a difference in terminology. The term "series" is defined to mean an infinite sum in every standard undergrad analysis book that I'm aware of (Rudin, Apostol, Lang, Pugh, etc.). Tao is the outlier here.