Can an infinite series be thought of as adding up “infinitely many” terms?Why is an infinite series not considered an infinite sum of terms?Geometric Series and its infinite sumWhy is an infinite series not considered an infinite sum of terms?sum of an infinite geometric seriesWhen adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?Series convergence, numerical termsExample of two converging series where partial sums change orders infinitely oftenConfusion around the way infinite series are defined, and how they are usedWhy does the concept of limits necessitate that infinite series can be assigned a value?
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Can an infinite series be thought of as adding up “infinitely many” terms?
Why is an infinite series not considered an infinite sum of terms?Geometric Series and its infinite sumWhy is an infinite series not considered an infinite sum of terms?sum of an infinite geometric seriesWhen adding or subtracting two infinite sums, why is there no issue with “staggering” or arbitrarily manipulating the “alignment” of terms?Series convergence, numerical termsExample of two converging series where partial sums change orders infinitely oftenConfusion around the way infinite series are defined, and how they are usedWhy does the concept of limits necessitate that infinite series can be assigned a value?
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$begingroup$
Formally, I understand that infinite series are not defined by adding up "infinitely many" terms, but are instead defined as equalling their limit. As user Brian M. Scott outlined in an answer to a similar question (Why is an infinite series not considered an infinite sum of terms?), it is easier to define infinite series by the limit of their partial sums than by considering every single term. However, one geometric proof of the convergence of $1 + 1/2 + 1/4 + cdots$ has made me question why we circumvent the problem of adding infinitely many terms with the concept of the limit:
Image credit: https://www.mathsisfun.com/algebra/infinite-series.html
When you look at the diagram above, it seems like every single term has been included (not literally, but it is clear what the diagram represents). Furthermore, if you plotted the above shape on the Cartesian plane, the area of the shape would be $1$. Even the point $(0.99999,0.99999)$ would be covered by a square/rectangle. When all of the terms have been plotted, it seems like the infinite series not only tends to 1, it equals $1$. To me, this is not just because we define an infinite series to equal its limit. The limit only concerns the partial sums, whereas the diagram shows that even if we allow ourselves to add infinitely many terms, there is no immediate contradiction. Obviously, defining infinite series like this formally can create problems: infinite series do not have always have the commutative property, for example. However, is there anything conceptually wrong with thinking of infinite series as adding up infinitely many terms, even though technically this can lead to some problems?
sequences-and-series definition intuition
asked 8 hours ago
JoeJoe
3651 silver badge10 bronze badges
$endgroup$
|
show 1 more comment
$begingroup$
Formally, I understand that infinite series are not defined by adding up "infinitely many" terms, but are instead defined as equalling their limit. As user Brian M. Scott outlined in an answer to a similar question (Why is an infinite series not considered an infinite sum of terms?), it is easier to define infinite series by the limit of their partial sums than by considering every single term. However, one geometric proof of the convergence of $1 + 1/2 + 1/4 + cdots$ has made me question why we circumvent the problem of adding infinitely many terms with the concept of the limit:
Image credit: https://www.mathsisfun.com/algebra/infinite-series.html
When you look at the diagram above, it seems like every single term has been included (not literally, but it is clear what the diagram represents). Furthermore, if you plotted the above shape on the Cartesian plane, the area of the shape would be $1$. Even the point $(0.99999,0.99999)$ would be covered by a square/rectangle. When all of the terms have been plotted, it seems like the infinite series not only tends to 1, it equals $1$. To me, this is not just because we define an infinite series to equal its limit. The limit only concerns the partial sums, whereas the diagram shows that even if we allow ourselves to add infinitely many terms, there is no immediate contradiction. Obviously, defining infinite series like this formally can create problems: infinite series do not have always have the commutative property, for example. However, is there anything conceptually wrong with thinking of infinite series as adding up infinitely many terms, even though technically this can lead to some problems?
sequences-and-series definition intuition
asked 8 hours ago
JoeJoe
3651 silver badge10 bronze badges
$endgroup$
$begingroup$
An infinite sum is the sum of an infinite number of things. We use limits because we can't add them two at a time in a finite amount of time. Do a search for Zeno's paradox. In all examples the "paradox" comes from assuming that an infinite sum is infinite.
$endgroup$
– John Douma
8 hours ago
1
$begingroup$
@JohnDouma if you add the first and second in $2^-2$ seconds, then add the third to this in $2^-3$ seconds, the fourth in $2^-4$ seconds, and so on, then each step will have taken a positive amount of time and you will have added them all up in half a second
$endgroup$
– Henry
8 hours ago
1
$begingroup$
@JohnDouma Thanks for answering. So we never try to define infinite series directly because continually adding the pairs of numbers gets us nowhere. Instead, we use limits to say however many terms we add the series will never get higher than this point. But does it make sense to say after adding up "infinitely many" terms the series equals its limit? After all, I thought the resolution to Zeno's Paradox was that you can add up infinitely many terms even in a finite time space.
$endgroup$
– Joe
8 hours ago
$begingroup$
@Henry What time will you add the last two?
$endgroup$
– John Douma
8 hours ago
$begingroup$
@JohnDouma Obviously there is no "last two" in an infinite series, but you will finish the additions precisely half a second after you start
$endgroup$
– Henry
2 hours ago
|
show 1 more comment
$begingroup$
Formally, I understand that infinite series are not defined by adding up "infinitely many" terms, but are instead defined as equalling their limit. As user Brian M. Scott outlined in an answer to a similar question (Why is an infinite series not considered an infinite sum of terms?), it is easier to define infinite series by the limit of their partial sums than by considering every single term. However, one geometric proof of the convergence of $1 + 1/2 + 1/4 + cdots$ has made me question why we circumvent the problem of adding infinitely many terms with the concept of the limit:
Image credit: https://www.mathsisfun.com/algebra/infinite-series.html
When you look at the diagram above, it seems like every single term has been included (not literally, but it is clear what the diagram represents). Furthermore, if you plotted the above shape on the Cartesian plane, the area of the shape would be $1$. Even the point $(0.99999,0.99999)$ would be covered by a square/rectangle. When all of the terms have been plotted, it seems like the infinite series not only tends to 1, it equals $1$. To me, this is not just because we define an infinite series to equal its limit. The limit only concerns the partial sums, whereas the diagram shows that even if we allow ourselves to add infinitely many terms, there is no immediate contradiction. Obviously, defining infinite series like this formally can create problems: infinite series do not have always have the commutative property, for example. However, is there anything conceptually wrong with thinking of infinite series as adding up infinitely many terms, even though technically this can lead to some problems?
sequences-and-series definition intuition
asked 8 hours ago
JoeJoe
3651 silver badge10 bronze badges
$endgroup$
Formally, I understand that infinite series are not defined by adding up "infinitely many" terms, but are instead defined as equalling their limit. As user Brian M. Scott outlined in an answer to a similar question (Why is an infinite series not considered an infinite sum of terms?), it is easier to define infinite series by the limit of their partial sums than by considering every single term. However, one geometric proof of the convergence of $1 + 1/2 + 1/4 + cdots$ has made me question why we circumvent the problem of adding infinitely many terms with the concept of the limit:
Image credit: https://www.mathsisfun.com/algebra/infinite-series.html
When you look at the diagram above, it seems like every single term has been included (not literally, but it is clear what the diagram represents). Furthermore, if you plotted the above shape on the Cartesian plane, the area of the shape would be $1$. Even the point $(0.99999,0.99999)$ would be covered by a square/rectangle. When all of the terms have been plotted, it seems like the infinite series not only tends to 1, it equals $1$. To me, this is not just because we define an infinite series to equal its limit. The limit only concerns the partial sums, whereas the diagram shows that even if we allow ourselves to add infinitely many terms, there is no immediate contradiction. Obviously, defining infinite series like this formally can create problems: infinite series do not have always have the commutative property, for example. However, is there anything conceptually wrong with thinking of infinite series as adding up infinitely many terms, even though technically this can lead to some problems?
sequences-and-series definition intuition
sequences-and-series definition intuition
asked 8 hours ago
JoeJoe
3651 silver badge10 bronze badges
asked 8 hours ago
JoeJoe
3651 silver badge10 bronze badges
edited 8 hours ago
Joe
asked 8 hours ago
JoeJoe
3651 silver badge10 bronze badges
asked 8 hours ago
JoeJoe
3651 silver badge10 bronze badges
asked 8 hours ago
JoeJoe
3651 silver badge10 bronze badges
3651 silver badge10 bronze badges
$begingroup$
An infinite sum is the sum of an infinite number of things. We use limits because we can't add them two at a time in a finite amount of time. Do a search for Zeno's paradox. In all examples the "paradox" comes from assuming that an infinite sum is infinite.
$endgroup$
– John Douma
8 hours ago
1
$begingroup$
@JohnDouma if you add the first and second in $2^-2$ seconds, then add the third to this in $2^-3$ seconds, the fourth in $2^-4$ seconds, and so on, then each step will have taken a positive amount of time and you will have added them all up in half a second
$endgroup$
– Henry
8 hours ago
1
$begingroup$
@JohnDouma Thanks for answering. So we never try to define infinite series directly because continually adding the pairs of numbers gets us nowhere. Instead, we use limits to say however many terms we add the series will never get higher than this point. But does it make sense to say after adding up "infinitely many" terms the series equals its limit? After all, I thought the resolution to Zeno's Paradox was that you can add up infinitely many terms even in a finite time space.
$endgroup$
– Joe
8 hours ago
$begingroup$
@Henry What time will you add the last two?
$endgroup$
– John Douma
8 hours ago
$begingroup$
@JohnDouma Obviously there is no "last two" in an infinite series, but you will finish the additions precisely half a second after you start
$endgroup$
– Henry
2 hours ago
|
show 1 more comment
$begingroup$
An infinite sum is the sum of an infinite number of things. We use limits because we can't add them two at a time in a finite amount of time. Do a search for Zeno's paradox. In all examples the "paradox" comes from assuming that an infinite sum is infinite.
$endgroup$
– John Douma
8 hours ago
1
$begingroup$
@JohnDouma if you add the first and second in $2^-2$ seconds, then add the third to this in $2^-3$ seconds, the fourth in $2^-4$ seconds, and so on, then each step will have taken a positive amount of time and you will have added them all up in half a second
$endgroup$
– Henry
8 hours ago
1
$begingroup$
@JohnDouma Thanks for answering. So we never try to define infinite series directly because continually adding the pairs of numbers gets us nowhere. Instead, we use limits to say however many terms we add the series will never get higher than this point. But does it make sense to say after adding up "infinitely many" terms the series equals its limit? After all, I thought the resolution to Zeno's Paradox was that you can add up infinitely many terms even in a finite time space.
$endgroup$
– Joe
8 hours ago
$begingroup$
@Henry What time will you add the last two?
$endgroup$
– John Douma
8 hours ago
$begingroup$
@JohnDouma Obviously there is no "last two" in an infinite series, but you will finish the additions precisely half a second after you start
$endgroup$
– Henry
2 hours ago
$begingroup$
An infinite sum is the sum of an infinite number of things. We use limits because we can't add them two at a time in a finite amount of time. Do a search for Zeno's paradox. In all examples the "paradox" comes from assuming that an infinite sum is infinite.
$endgroup$
– John Douma
8 hours ago
$begingroup$
An infinite sum is the sum of an infinite number of things. We use limits because we can't add them two at a time in a finite amount of time. Do a search for Zeno's paradox. In all examples the "paradox" comes from assuming that an infinite sum is infinite.
$endgroup$
– John Douma
8 hours ago
1
1
$begingroup$
@JohnDouma if you add the first and second in $2^-2$ seconds, then add the third to this in $2^-3$ seconds, the fourth in $2^-4$ seconds, and so on, then each step will have taken a positive amount of time and you will have added them all up in half a second
$endgroup$
– Henry
8 hours ago
$begingroup$
@JohnDouma if you add the first and second in $2^-2$ seconds, then add the third to this in $2^-3$ seconds, the fourth in $2^-4$ seconds, and so on, then each step will have taken a positive amount of time and you will have added them all up in half a second
$endgroup$
– Henry
8 hours ago
1
1
$begingroup$
@JohnDouma Thanks for answering. So we never try to define infinite series directly because continually adding the pairs of numbers gets us nowhere. Instead, we use limits to say however many terms we add the series will never get higher than this point. But does it make sense to say after adding up "infinitely many" terms the series equals its limit? After all, I thought the resolution to Zeno's Paradox was that you can add up infinitely many terms even in a finite time space.
$endgroup$
– Joe
8 hours ago
$begingroup$
@JohnDouma Thanks for answering. So we never try to define infinite series directly because continually adding the pairs of numbers gets us nowhere. Instead, we use limits to say however many terms we add the series will never get higher than this point. But does it make sense to say after adding up "infinitely many" terms the series equals its limit? After all, I thought the resolution to Zeno's Paradox was that you can add up infinitely many terms even in a finite time space.
$endgroup$
– Joe
8 hours ago
$begingroup$
@Henry What time will you add the last two?
$endgroup$
– John Douma
8 hours ago
$begingroup$
@Henry What time will you add the last two?
$endgroup$
– John Douma
8 hours ago
$begingroup$
@JohnDouma Obviously there is no "last two" in an infinite series, but you will finish the additions precisely half a second after you start
$endgroup$
– Henry
2 hours ago
$begingroup$
@JohnDouma Obviously there is no "last two" in an infinite series, but you will finish the additions precisely half a second after you start
$endgroup$
– Henry
2 hours ago
|
show 1 more comment
3 Answers
3
active
oldest
votes
$begingroup$
The answer to your question is essentially here:
Obviously, defining infinite series like this formally can create
problems
It is intuitively OK to think of summing an infinite series as
adding up "infinitely many" terms
but if you want to prove theorems about those sums and avoid the problems you need a formal definition. Mathematicians have discovered that a good way to do that is to replace "adding infinitely many by numbers" by proving infinitely many statements. That's what the "for every $epsilon$" does in the formal definition of limits.
answered 8 hours ago
Ethan BolkerEthan Bolker
55.7k5 gold badges63 silver badges133 bronze badges
$endgroup$
$begingroup$
Thanks, this was a very insightful answer. As a follow-up question: in the resolution of Achilles and the Tortoise, do we (in this specific case) allow ourselves to do infinitely many steps (and in effect add every single term up) because each step takes less and less time too?
$endgroup$
– Joe
8 hours ago
$begingroup$
@Joe I don't think so. Achilles and the Tortoise is a linguistic paradox and perhaps a philosophical paradox but not a mathematical one. Mathematicians just define limits formally and reason about them when so defined. That definition produces results that are consistent with intuition, free of contradictions (though perhaps not provably so) and useful. That's enough to get on with the job.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment
|
$begingroup$
It is possible to interpret series as a sum of infinitely many of things.
(I don't know the right way enough to give you a short answer, see ref in (4) instead).However, to do that correctly, you need to define infinitely large and
infinitesimal rigorously. This is possible only after Abraham Robinson develop
his Non-standard analysis in early $1960s$.Everything you learn about infinity in University are developed way before the time we know how to handle infinity rigorously. If you mix the incomplete concept of infinity you have learn with the $epsilon-delta$ approach of analysis, you are making your life a misery.
Since Abraham's work, there are other people who attempt to redevelop calculus
using infinitesimal approach that are more accessible to students. If you are really interested in this sort of stuff, my personal recommendation is H. Jerome Keisler's Elementary Calculus: an infinitesimal approach. This book is quite readable.
answered 7 hours ago
achille huiachille hui
99.8k5 gold badges136 silver badges271 bronze badges
$endgroup$
$begingroup$
+1 I thought of appending something along these lines to my answer. You said it much better than I would have.
$endgroup$
– Ethan Bolker
7 hours ago
add a comment
|
$begingroup$
I compare it to proving that the natural numbers have as many elements as the positive rationals. You can demonstrate a bijection between them with enough detail that you can tell what the millionth term in the sequence in or what is the index of 335/113, but we need to come to terms with the notion that "counting to infinity" is something that we can only consider under the terms with which we have defined those words.
The same thing is happening here. I could ask for any point on that square except the upper-right corner, and you'd be able to tell me the label of the unique rectangle that includes it. But you're still only drawing a finite number of those rectangles and you still can't fill in that upper-right corner. To respect those important points, we say that the limit of that series is 1 instead of saying that the sum is 1.
answered 8 hours ago
Matthew DalyMatthew Daly
4,9621 gold badge7 silver badges27 bronze badges
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add a comment
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The answer to your question is essentially here:
Obviously, defining infinite series like this formally can create
problems
It is intuitively OK to think of summing an infinite series as
adding up "infinitely many" terms
but if you want to prove theorems about those sums and avoid the problems you need a formal definition. Mathematicians have discovered that a good way to do that is to replace "adding infinitely many by numbers" by proving infinitely many statements. That's what the "for every $epsilon$" does in the formal definition of limits.
answered 8 hours ago
Ethan BolkerEthan Bolker
55.7k5 gold badges63 silver badges133 bronze badges
$endgroup$
$begingroup$
Thanks, this was a very insightful answer. As a follow-up question: in the resolution of Achilles and the Tortoise, do we (in this specific case) allow ourselves to do infinitely many steps (and in effect add every single term up) because each step takes less and less time too?
$endgroup$
– Joe
8 hours ago
$begingroup$
@Joe I don't think so. Achilles and the Tortoise is a linguistic paradox and perhaps a philosophical paradox but not a mathematical one. Mathematicians just define limits formally and reason about them when so defined. That definition produces results that are consistent with intuition, free of contradictions (though perhaps not provably so) and useful. That's enough to get on with the job.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment
|
$begingroup$
The answer to your question is essentially here:
Obviously, defining infinite series like this formally can create
problems
It is intuitively OK to think of summing an infinite series as
adding up "infinitely many" terms
but if you want to prove theorems about those sums and avoid the problems you need a formal definition. Mathematicians have discovered that a good way to do that is to replace "adding infinitely many by numbers" by proving infinitely many statements. That's what the "for every $epsilon$" does in the formal definition of limits.
answered 8 hours ago
Ethan BolkerEthan Bolker
55.7k5 gold badges63 silver badges133 bronze badges
$endgroup$
$begingroup$
Thanks, this was a very insightful answer. As a follow-up question: in the resolution of Achilles and the Tortoise, do we (in this specific case) allow ourselves to do infinitely many steps (and in effect add every single term up) because each step takes less and less time too?
$endgroup$
– Joe
8 hours ago
$begingroup$
@Joe I don't think so. Achilles and the Tortoise is a linguistic paradox and perhaps a philosophical paradox but not a mathematical one. Mathematicians just define limits formally and reason about them when so defined. That definition produces results that are consistent with intuition, free of contradictions (though perhaps not provably so) and useful. That's enough to get on with the job.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment
|
$begingroup$
The answer to your question is essentially here:
Obviously, defining infinite series like this formally can create
problems
It is intuitively OK to think of summing an infinite series as
adding up "infinitely many" terms
but if you want to prove theorems about those sums and avoid the problems you need a formal definition. Mathematicians have discovered that a good way to do that is to replace "adding infinitely many by numbers" by proving infinitely many statements. That's what the "for every $epsilon$" does in the formal definition of limits.
answered 8 hours ago
Ethan BolkerEthan Bolker
55.7k5 gold badges63 silver badges133 bronze badges
$endgroup$
The answer to your question is essentially here:
Obviously, defining infinite series like this formally can create
problems
It is intuitively OK to think of summing an infinite series as
adding up "infinitely many" terms
but if you want to prove theorems about those sums and avoid the problems you need a formal definition. Mathematicians have discovered that a good way to do that is to replace "adding infinitely many by numbers" by proving infinitely many statements. That's what the "for every $epsilon$" does in the formal definition of limits.
answered 8 hours ago
Ethan BolkerEthan Bolker
55.7k5 gold badges63 silver badges133 bronze badges
answered 8 hours ago
Ethan BolkerEthan Bolker
55.7k5 gold badges63 silver badges133 bronze badges
answered 8 hours ago
Ethan BolkerEthan Bolker
55.7k5 gold badges63 silver badges133 bronze badges
answered 8 hours ago
Ethan BolkerEthan Bolker
55.7k5 gold badges63 silver badges133 bronze badges
55.7k5 gold badges63 silver badges133 bronze badges
$begingroup$
Thanks, this was a very insightful answer. As a follow-up question: in the resolution of Achilles and the Tortoise, do we (in this specific case) allow ourselves to do infinitely many steps (and in effect add every single term up) because each step takes less and less time too?
$endgroup$
– Joe
8 hours ago
$begingroup$
@Joe I don't think so. Achilles and the Tortoise is a linguistic paradox and perhaps a philosophical paradox but not a mathematical one. Mathematicians just define limits formally and reason about them when so defined. That definition produces results that are consistent with intuition, free of contradictions (though perhaps not provably so) and useful. That's enough to get on with the job.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment
|
$begingroup$
Thanks, this was a very insightful answer. As a follow-up question: in the resolution of Achilles and the Tortoise, do we (in this specific case) allow ourselves to do infinitely many steps (and in effect add every single term up) because each step takes less and less time too?
$endgroup$
– Joe
8 hours ago
$begingroup$
@Joe I don't think so. Achilles and the Tortoise is a linguistic paradox and perhaps a philosophical paradox but not a mathematical one. Mathematicians just define limits formally and reason about them when so defined. That definition produces results that are consistent with intuition, free of contradictions (though perhaps not provably so) and useful. That's enough to get on with the job.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
Thanks, this was a very insightful answer. As a follow-up question: in the resolution of Achilles and the Tortoise, do we (in this specific case) allow ourselves to do infinitely many steps (and in effect add every single term up) because each step takes less and less time too?
$endgroup$
– Joe
8 hours ago
$begingroup$
Thanks, this was a very insightful answer. As a follow-up question: in the resolution of Achilles and the Tortoise, do we (in this specific case) allow ourselves to do infinitely many steps (and in effect add every single term up) because each step takes less and less time too?
$endgroup$
– Joe
8 hours ago
$begingroup$
@Joe I don't think so. Achilles and the Tortoise is a linguistic paradox and perhaps a philosophical paradox but not a mathematical one. Mathematicians just define limits formally and reason about them when so defined. That definition produces results that are consistent with intuition, free of contradictions (though perhaps not provably so) and useful. That's enough to get on with the job.
$endgroup$
– Ethan Bolker
8 hours ago
$begingroup$
@Joe I don't think so. Achilles and the Tortoise is a linguistic paradox and perhaps a philosophical paradox but not a mathematical one. Mathematicians just define limits formally and reason about them when so defined. That definition produces results that are consistent with intuition, free of contradictions (though perhaps not provably so) and useful. That's enough to get on with the job.
$endgroup$
– Ethan Bolker
8 hours ago
add a comment
|
$begingroup$
It is possible to interpret series as a sum of infinitely many of things.
(I don't know the right way enough to give you a short answer, see ref in (4) instead).However, to do that correctly, you need to define infinitely large and
infinitesimal rigorously. This is possible only after Abraham Robinson develop
his Non-standard analysis in early $1960s$.Everything you learn about infinity in University are developed way before the time we know how to handle infinity rigorously. If you mix the incomplete concept of infinity you have learn with the $epsilon-delta$ approach of analysis, you are making your life a misery.
Since Abraham's work, there are other people who attempt to redevelop calculus
using infinitesimal approach that are more accessible to students. If you are really interested in this sort of stuff, my personal recommendation is H. Jerome Keisler's Elementary Calculus: an infinitesimal approach. This book is quite readable.
answered 7 hours ago
achille huiachille hui
99.8k5 gold badges136 silver badges271 bronze badges
$endgroup$
$begingroup$
+1 I thought of appending something along these lines to my answer. You said it much better than I would have.
$endgroup$
– Ethan Bolker
7 hours ago
add a comment
|
$begingroup$
It is possible to interpret series as a sum of infinitely many of things.
(I don't know the right way enough to give you a short answer, see ref in (4) instead).However, to do that correctly, you need to define infinitely large and
infinitesimal rigorously. This is possible only after Abraham Robinson develop
his Non-standard analysis in early $1960s$.Everything you learn about infinity in University are developed way before the time we know how to handle infinity rigorously. If you mix the incomplete concept of infinity you have learn with the $epsilon-delta$ approach of analysis, you are making your life a misery.
Since Abraham's work, there are other people who attempt to redevelop calculus
using infinitesimal approach that are more accessible to students. If you are really interested in this sort of stuff, my personal recommendation is H. Jerome Keisler's Elementary Calculus: an infinitesimal approach. This book is quite readable.
answered 7 hours ago
achille huiachille hui
99.8k5 gold badges136 silver badges271 bronze badges
$endgroup$
$begingroup$
+1 I thought of appending something along these lines to my answer. You said it much better than I would have.
$endgroup$
– Ethan Bolker
7 hours ago
add a comment
|
$begingroup$
It is possible to interpret series as a sum of infinitely many of things.
(I don't know the right way enough to give you a short answer, see ref in (4) instead).However, to do that correctly, you need to define infinitely large and
infinitesimal rigorously. This is possible only after Abraham Robinson develop
his Non-standard analysis in early $1960s$.Everything you learn about infinity in University are developed way before the time we know how to handle infinity rigorously. If you mix the incomplete concept of infinity you have learn with the $epsilon-delta$ approach of analysis, you are making your life a misery.
Since Abraham's work, there are other people who attempt to redevelop calculus
using infinitesimal approach that are more accessible to students. If you are really interested in this sort of stuff, my personal recommendation is H. Jerome Keisler's Elementary Calculus: an infinitesimal approach. This book is quite readable.
answered 7 hours ago
achille huiachille hui
99.8k5 gold badges136 silver badges271 bronze badges
$endgroup$
It is possible to interpret series as a sum of infinitely many of things.
(I don't know the right way enough to give you a short answer, see ref in (4) instead).However, to do that correctly, you need to define infinitely large and
infinitesimal rigorously. This is possible only after Abraham Robinson develop
his Non-standard analysis in early $1960s$.Everything you learn about infinity in University are developed way before the time we know how to handle infinity rigorously. If you mix the incomplete concept of infinity you have learn with the $epsilon-delta$ approach of analysis, you are making your life a misery.
Since Abraham's work, there are other people who attempt to redevelop calculus
using infinitesimal approach that are more accessible to students. If you are really interested in this sort of stuff, my personal recommendation is H. Jerome Keisler's Elementary Calculus: an infinitesimal approach. This book is quite readable.
answered 7 hours ago
achille huiachille hui
99.8k5 gold badges136 silver badges271 bronze badges
answered 7 hours ago
achille huiachille hui
99.8k5 gold badges136 silver badges271 bronze badges
answered 7 hours ago
achille huiachille hui
99.8k5 gold badges136 silver badges271 bronze badges
answered 7 hours ago
achille huiachille hui
99.8k5 gold badges136 silver badges271 bronze badges
99.8k5 gold badges136 silver badges271 bronze badges
$begingroup$
+1 I thought of appending something along these lines to my answer. You said it much better than I would have.
$endgroup$
– Ethan Bolker
7 hours ago
add a comment
|
$begingroup$
+1 I thought of appending something along these lines to my answer. You said it much better than I would have.
$endgroup$
– Ethan Bolker
7 hours ago
$begingroup$
+1 I thought of appending something along these lines to my answer. You said it much better than I would have.
$endgroup$
– Ethan Bolker
7 hours ago
$begingroup$
+1 I thought of appending something along these lines to my answer. You said it much better than I would have.
$endgroup$
– Ethan Bolker
7 hours ago
add a comment
|
$begingroup$
I compare it to proving that the natural numbers have as many elements as the positive rationals. You can demonstrate a bijection between them with enough detail that you can tell what the millionth term in the sequence in or what is the index of 335/113, but we need to come to terms with the notion that "counting to infinity" is something that we can only consider under the terms with which we have defined those words.
The same thing is happening here. I could ask for any point on that square except the upper-right corner, and you'd be able to tell me the label of the unique rectangle that includes it. But you're still only drawing a finite number of those rectangles and you still can't fill in that upper-right corner. To respect those important points, we say that the limit of that series is 1 instead of saying that the sum is 1.
answered 8 hours ago
Matthew DalyMatthew Daly
4,9621 gold badge7 silver badges27 bronze badges
$endgroup$
add a comment
|
$begingroup$
I compare it to proving that the natural numbers have as many elements as the positive rationals. You can demonstrate a bijection between them with enough detail that you can tell what the millionth term in the sequence in or what is the index of 335/113, but we need to come to terms with the notion that "counting to infinity" is something that we can only consider under the terms with which we have defined those words.
The same thing is happening here. I could ask for any point on that square except the upper-right corner, and you'd be able to tell me the label of the unique rectangle that includes it. But you're still only drawing a finite number of those rectangles and you still can't fill in that upper-right corner. To respect those important points, we say that the limit of that series is 1 instead of saying that the sum is 1.
answered 8 hours ago
Matthew DalyMatthew Daly
4,9621 gold badge7 silver badges27 bronze badges
$endgroup$
add a comment
|
$begingroup$
I compare it to proving that the natural numbers have as many elements as the positive rationals. You can demonstrate a bijection between them with enough detail that you can tell what the millionth term in the sequence in or what is the index of 335/113, but we need to come to terms with the notion that "counting to infinity" is something that we can only consider under the terms with which we have defined those words.
The same thing is happening here. I could ask for any point on that square except the upper-right corner, and you'd be able to tell me the label of the unique rectangle that includes it. But you're still only drawing a finite number of those rectangles and you still can't fill in that upper-right corner. To respect those important points, we say that the limit of that series is 1 instead of saying that the sum is 1.
answered 8 hours ago
Matthew DalyMatthew Daly
4,9621 gold badge7 silver badges27 bronze badges
$endgroup$
I compare it to proving that the natural numbers have as many elements as the positive rationals. You can demonstrate a bijection between them with enough detail that you can tell what the millionth term in the sequence in or what is the index of 335/113, but we need to come to terms with the notion that "counting to infinity" is something that we can only consider under the terms with which we have defined those words.
The same thing is happening here. I could ask for any point on that square except the upper-right corner, and you'd be able to tell me the label of the unique rectangle that includes it. But you're still only drawing a finite number of those rectangles and you still can't fill in that upper-right corner. To respect those important points, we say that the limit of that series is 1 instead of saying that the sum is 1.
answered 8 hours ago
Matthew DalyMatthew Daly
4,9621 gold badge7 silver badges27 bronze badges
answered 8 hours ago
Matthew DalyMatthew Daly
4,9621 gold badge7 silver badges27 bronze badges
answered 8 hours ago
Matthew DalyMatthew Daly
4,9621 gold badge7 silver badges27 bronze badges
answered 8 hours ago
Matthew DalyMatthew Daly
4,9621 gold badge7 silver badges27 bronze badges
4,9621 gold badge7 silver badges27 bronze badges
add a comment
|
add a comment
|
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An infinite sum is the sum of an infinite number of things. We use limits because we can't add them two at a time in a finite amount of time. Do a search for Zeno's paradox. In all examples the "paradox" comes from assuming that an infinite sum is infinite.
$endgroup$
– John Douma
8 hours ago
1
$begingroup$
@JohnDouma if you add the first and second in $2^-2$ seconds, then add the third to this in $2^-3$ seconds, the fourth in $2^-4$ seconds, and so on, then each step will have taken a positive amount of time and you will have added them all up in half a second
$endgroup$
– Henry
8 hours ago
1
$begingroup$
@JohnDouma Thanks for answering. So we never try to define infinite series directly because continually adding the pairs of numbers gets us nowhere. Instead, we use limits to say however many terms we add the series will never get higher than this point. But does it make sense to say after adding up "infinitely many" terms the series equals its limit? After all, I thought the resolution to Zeno's Paradox was that you can add up infinitely many terms even in a finite time space.
$endgroup$
– Joe
8 hours ago
$begingroup$
@Henry What time will you add the last two?
$endgroup$
– John Douma
8 hours ago
$begingroup$
@JohnDouma Obviously there is no "last two" in an infinite series, but you will finish the additions precisely half a second after you start
$endgroup$
– Henry
2 hours ago