absolutely the best teacher on the planet everyday I see your videos and I really appreciate your great work and help , thank you for your extraordinary contributing to math community I am from Iran and I am really greatful to live in this incredible time , that I can access this quality of learning thank you so much
When I first clicked on this video, I had no idea that it would lead to a discussion of absolute vs. conditional convergence. I've watched your other calculus videos about convergence and this explanation is great, even better than your other videos (which are also top notch). Thank you for providing another approach to this complex topic.
Really good. I've done a lot of math in my many years but your videos show there are always new and exciting things out there. I've been watching your videos, which appeal to intuition (with some rigor) and then seeking out rigorous proofs ... what a great way to learn. Thank you so much.
Great video! The examples at the beginning, followed by the rigorous discussion, all came together for a great and unique format of the lesson! I have seen before the argument that rearranging conditionally convergent series can cause them to converge to any value you want, and the argument makes sense. But it's just so TEDIOUS to think about it that way, since you have to alternate overshooting and undershooting the value forever, as you zero in on it! Such is the nature of infinite entities, I guess!
I think the reason this seems bizarre is exclusively due to how teachers present the topic to their students, which is more often than not, incorrect and confusing. Here is what people need to understand: series are not "infinite sums" in any meaningful way, and the only reason they "appear" to be an "infinite sum" is because we like to abuse our notation in order to simplify our lives, visually giving the impression that there is a "sum of infinitely many numbers" where there is none. We can talk about about the set of sequences f : N -> R, and then define an operator on this set of sequences, the operator S on this sequence, such that S[f(n)] = s(n), where s(0) = f(0), and s(n + 1) = s(n) + f(n + 1) for every n, and then one can talk about lim s(n) (n -> ♾), which is also an operator on this set of sequences, perhaps you can denote it the operator L. As such, an infinite series is nothing more than the value of L{S[f(n)]}, which can also be described by a single operator on sequences, the operator Series. So ultimately, the Series operator on any sequence is a function that takes a sequence as an input, and returns an output for some of those sequences. That is all it is. There is no "infinite sum" being carried out, we are simply transforming sequences into other sequences, and then taking the limit of those output sequences, and determining whether it exists or not. However, because expressing this concisely and rigorously can get very tedious, and it can be difficult to explain, we simply abuse the notation and write f(0) + f(1) + •••, which gives the impression that there are infinitely many summands. This is not the case at all. We are not evaluating any sum. We are evaluating a limit of an antidifference of a sequence, and it just so happens that addition symbols are used to conveniently notate this in abbreviated form whenever we decide to give up rigor. Once this is understood, the rearrangement theorems become quite intuitive: there is no "strangeness" being introduced to addition when accounting for infinity, because we are not actually doing infinite addition. We are doing limits of sequences, and realizing that, if you change a sequence, you may obtain a different limit, is actually rather trivial. The problem is that math teachers do a terrible job at teaching series correctly.
Maybe this is why it breaks my intuition. If you frontload a series so that it adds up to 100 right at the beginning, you're just borrowing those numbers from the future such that eventually they will have to be omitted, and that hundred is eventually corrected out. That is, if the series is truly infinite. But if the series is not truly infinite - i.e. it approaches infinity rather than reaching infinity (which of course can never happen) - then we can pretend it adds up to anything we want. Right?
@@danielweitsman3444 You are making the exact kind of mistake I highlighted within my own post and I warned against: you are thinking of the operation involved as being a *sum of numbers,* as corresponding to *addition.* A series is not at all a sum of numbers. The defining feature of what a series is equal to boils down to the _limit of a sequence._ The sequence itself is defined in terms of sums of terms of _another sequence,_ but the final step is, and always is, a limit. So the addition operation is completely irrelevant, actually. If you rearrange infinitely many terms of a sequence, then the limit of the sequence will change, generally speaking, regardless of how the sequence is defined.
@@angelmendez-rivera351 Turns out this is on the very next page after the bookmark in my Stewart Calculus book, so I just had to read one page ahead. However, it still seems like a radical idea. I can see how the rearrangement of an alternating series changes the geometry of the function such that it converges around a different number, but I'm unable to make the conceptual leap to where an infinite series is not the sum of never-ending addition operators. Is there some website that explains this graphically?
@@danielweitsman3444 I am not sure I understand what a graphical demonstration of this would look like, because ultimately, what I am talking about is just the definition of a series. Here is how I recommend thinking about it: you have a sequence α = (a0, a1, ...). The limit operator L is a machine, which takes the sequence α as the input, and spits out the real number lim α(n) (n -> ∞) as the output. There is also another operator, the series operator Series, which is another machine, whose input is the sequence α, and whose output is some real number Series(α). The difference here is that Series is a composite machine: it is made up of two parts. First, given the sequence α, you have to find a different sequence β, which is the sequence of partial sums of α. Then, you find L(β) = lim β(n) (n -> ∞). We define Series(α) = L(β). Here is a concrete example. Imagine α(n) = 1/2^n. What is β(n), in this case? Well, β(n) = 2 - 2/2^n. You can prove this pretty easily, because β(0) = 0, and β(n + 1) - β(n) = α(n). So, Series(α) = L(β), and by definition, L(β) = lim 2 - 2/2^n (n -> ∞) = 2. Therefore, Series(α) = 2. However, the *traditional* notation for this, unfortunately, is 1 + 1/2 + 1/4 + 1/8 + ••• = 2. Because of this notation, people are misled to think that we are dealing with addition, even though, as I just explained it, the definition has little to do with addition in actuality. To be clear, the definition of β with respect to α does involve addition, simply because of the equation that β(n + 1) - β(n) = α(n). That being the said, it is involved in a very specific way that has nothing to do with just adding infinitely many numbers. Instead, I think it is more helpful to think of β as being the *discrete antiderivative* of the sequence α, the discrete analogue of an antiderivative for a function g of real numbers. It makes sense, because the operation β(n + 1) - β(n) is the discrete analogue of the derivative, and so, the equation β(n + 1) - β(n) = α(n) is the discrete analogue of the differential equation df/dx = g. Computing Series(α) is then just completely analogous to finding the (improper) integral from 0 to ∞ of g(x). This is a clearer way to understand what is happening. The definition of the discrete antiderivative does involve addition, yes, but again, it is indirect, and so, merely saying "you are adding numbers" does not give you any real information as to what you are genuinely doing when computing a series, especially because the more important operation involved in the computation is the limit operator L, which is fairly unique in how it works. This is why calculus is just one course dedicated to this one operator. Hopefully, this explanation helps.
Excellent stuff! This helped me deepen my understanding of conditional/absolute convergence. Not that my calc 2 prof is gonna quiz me on this later this week haha but it's certainly helps the concepts settle in
Wouldve been nice to see this closely after i had to prove/show that this was the case, to answer a question that without him mentioning it diredtly was solved. My argument was pretty much the one in the vid was fun to do so.(But i forgot/got confused about the conditional convergence and thought about that this wouldnt be true because if the terms get bigger than you cant ensure you can actually reach any number that easaly anymore, but from the monton decreasing value nature of the series that is quite obvious, to me now, was also define that way just forgot)
I think there is an issue at min 15: absolute convergence is achieved if and only if positive AND negative parts converge (it would be interesting to insert the demo by the way it is a very direct and easy one). So if series conditionnally converge then positive OR negative part diverges. Many thanks for sharing ressources, impressive work! Best
"absolute convergence is achieved if and only if positive AND negative parts converge" This is correct. "So if series conditionnally converge then positive OR negative part diverges." That is incorrect. If a series conditionally converges, both the positive AND negative parts _must_ diverge. Conditionally convergent is not the complement of absolutely convergent. There is a third case: divergent. If only the positive part diverges, then the whole series diverges to infinity. If only the negative part diverges, then the whole series diverges to negative infinity. Note that it is possible for a divergent series to have both the positive and negative parts diverge, so "both the positive and negative parts of the series diverge" is necessary for conditionally convergent, but not sufficient.
The typical strategy to rearrange any conditionally convergent series so you get a sum of M is to keep adding positive terms until you get above M, then keep adding negative terms until you get below M, and repeat this process. To get it to diverge to infinity, you can't use such a simple process as "keep adding positive terms until you get past infinity" because that will never happen, so you won't include any negative terms at all. Instead, what you can do is something like this: Keep adding positive terms until you get past 1. Then keep adding negative terms until you drop down to 0. Then keep adding positive terms until you get past 2. The keep adding negative terms until you drop down to 1. Then keep adding positive terms until you get past 3. Then keep adding negative terms until you drop down to 2. etc. This method will lead to divergence to infinity.
I think it's important to mention that convergence is likely only relevant because we have physics that shields us from infinity and allows us to have information and identity, things like the speed of light and the Plank length. If you really want to gain a deeper understanding of this kind of math you'll have to explore that kind of physics I think.
This is the most underatted channel ever, you literially saved my entire class's life. I cant thank you enough
Glad to hear it!
Love that you present a core difficult concept from Real Analysis in a way a kid could appreciate it. 👍
absolutely
the best teacher on the planet
everyday I see your videos and I really appreciate your great work and help ,
thank you for your extraordinary contributing to math community
I am from Iran and I am really greatful to live in this incredible time , that I can access this quality of learning
thank you so much
When I first clicked on this video, I had no idea that it would lead to a discussion of absolute vs. conditional convergence. I've watched your other calculus videos about convergence and this explanation is great, even better than your other videos (which are also top notch). Thank you for providing another approach to this complex topic.
I have seen this video many times, and I absolutely love it.
.. U honestly deserve many more subscribers.... I discovered ur channel while studying linear algebra... And I've also leant a lot more
Learnt*
Really good. I've done a lot of math in my many years but your videos show there are always new and exciting things out there. I've been watching your videos, which appeal to intuition (with some rigor) and then seeking out rigorous proofs ... what a great way to learn. Thank you so much.
Well Explained! Great Job!
Great video! The examples at the beginning, followed by the rigorous discussion, all came together for a great and unique format of the lesson!
I have seen before the argument that rearranging conditionally convergent series can cause them to converge to any value you want, and the argument makes sense. But it's just so TEDIOUS to think about it that way, since you have to alternate overshooting and undershooting the value forever, as you zero in on it! Such is the nature of infinite entities, I guess!
I am here after listening to the Bob Murphy podcast with Steve Patterson.
Link in the description is not valid anymore. I like your series, I am about to be a calculus master in 1 month :)
Amazing explanation. Congrats!
Excellent stuff! Reminds me of the grandi sum and zorns lemma
I think the reason this seems bizarre is exclusively due to how teachers present the topic to their students, which is more often than not, incorrect and confusing. Here is what people need to understand: series are not "infinite sums" in any meaningful way, and the only reason they "appear" to be an "infinite sum" is because we like to abuse our notation in order to simplify our lives, visually giving the impression that there is a "sum of infinitely many numbers" where there is none.
We can talk about about the set of sequences f : N -> R, and then define an operator on this set of sequences, the operator S on this sequence, such that S[f(n)] = s(n), where s(0) = f(0), and s(n + 1) = s(n) + f(n + 1) for every n, and then one can talk about lim s(n) (n -> ♾), which is also an operator on this set of sequences, perhaps you can denote it the operator L. As such, an infinite series is nothing more than the value of L{S[f(n)]}, which can also be described by a single operator on sequences, the operator Series. So ultimately, the Series operator on any sequence is a function that takes a sequence as an input, and returns an output for some of those sequences. That is all it is. There is no "infinite sum" being carried out, we are simply transforming sequences into other sequences, and then taking the limit of those output sequences, and determining whether it exists or not. However, because expressing this concisely and rigorously can get very tedious, and it can be difficult to explain, we simply abuse the notation and write f(0) + f(1) + •••, which gives the impression that there are infinitely many summands. This is not the case at all. We are not evaluating any sum. We are evaluating a limit of an antidifference of a sequence, and it just so happens that addition symbols are used to conveniently notate this in abbreviated form whenever we decide to give up rigor.
Once this is understood, the rearrangement theorems become quite intuitive: there is no "strangeness" being introduced to addition when accounting for infinity, because we are not actually doing infinite addition. We are doing limits of sequences, and realizing that, if you change a sequence, you may obtain a different limit, is actually rather trivial. The problem is that math teachers do a terrible job at teaching series correctly.
WOAH! your comments are always long.
Maybe this is why it breaks my intuition. If you frontload a series so that it adds up to 100 right at the beginning, you're just borrowing those numbers from the future such that eventually they will have to be omitted, and that hundred is eventually corrected out. That is, if the series is truly infinite. But if the series is not truly infinite - i.e. it approaches infinity rather than reaching infinity (which of course can never happen) - then we can pretend it adds up to anything we want. Right?
@@danielweitsman3444 You are making the exact kind of mistake I highlighted within my own post and I warned against: you are thinking of the operation involved as being a *sum of numbers,* as corresponding to *addition.* A series is not at all a sum of numbers. The defining feature of what a series is equal to boils down to the _limit of a sequence._ The sequence itself is defined in terms of sums of terms of _another sequence,_ but the final step is, and always is, a limit. So the addition operation is completely irrelevant, actually. If you rearrange infinitely many terms of a sequence, then the limit of the sequence will change, generally speaking, regardless of how the sequence is defined.
@@angelmendez-rivera351 Turns out this is on the very next page after the bookmark in my Stewart Calculus book, so I just had to read one page ahead. However, it still seems like a radical idea. I can see how the rearrangement of an alternating series changes the geometry of the function such that it converges around a different number, but I'm unable to make the conceptual leap to where an infinite series is not the sum of never-ending addition operators. Is there some website that explains this graphically?
@@danielweitsman3444 I am not sure I understand what a graphical demonstration of this would look like, because ultimately, what I am talking about is just the definition of a series.
Here is how I recommend thinking about it: you have a sequence α = (a0, a1, ...). The limit operator L is a machine, which takes the sequence α as the input, and spits out the real number lim α(n) (n -> ∞) as the output. There is also another operator, the series operator Series, which is another machine, whose input is the sequence α, and whose output is some real number Series(α). The difference here is that Series is a composite machine: it is made up of two parts. First, given the sequence α, you have to find a different sequence β, which is the sequence of partial sums of α. Then, you find L(β) = lim β(n) (n -> ∞). We define Series(α) = L(β).
Here is a concrete example. Imagine α(n) = 1/2^n. What is β(n), in this case? Well, β(n) = 2 - 2/2^n. You can prove this pretty easily, because β(0) = 0, and β(n + 1) - β(n) = α(n). So, Series(α) = L(β), and by definition, L(β) = lim 2 - 2/2^n (n -> ∞) = 2. Therefore, Series(α) = 2. However, the *traditional* notation for this, unfortunately, is 1 + 1/2 + 1/4 + 1/8 + ••• = 2. Because of this notation, people are misled to think that we are dealing with addition, even though, as I just explained it, the definition has little to do with addition in actuality.
To be clear, the definition of β with respect to α does involve addition, simply because of the equation that β(n + 1) - β(n) = α(n). That being the said, it is involved in a very specific way that has nothing to do with just adding infinitely many numbers. Instead, I think it is more helpful to think of β as being the *discrete antiderivative* of the sequence α, the discrete analogue of an antiderivative for a function g of real numbers. It makes sense, because the operation β(n + 1) - β(n) is the discrete analogue of the derivative, and so, the equation β(n + 1) - β(n) = α(n) is the discrete analogue of the differential equation df/dx = g. Computing Series(α) is then just completely analogous to finding the (improper) integral from 0 to ∞ of g(x). This is a clearer way to understand what is happening. The definition of the discrete antiderivative does involve addition, yes, but again, it is indirect, and so, merely saying "you are adding numbers" does not give you any real information as to what you are genuinely doing when computing a series, especially because the more important operation involved in the computation is the limit operator L, which is fairly unique in how it works. This is why calculus is just one course dedicated to this one operator. Hopefully, this explanation helps.
Practical approach, excellent.
Dead link for alternating harmonic series proof! Great stuff.
Excellent stuff! This helped me deepen my understanding of conditional/absolute convergence. Not that my calc 2 prof is gonna quiz me on this later this week haha but it's certainly helps the concepts settle in
So cool to understand maths like this...you can actually visualise the maths.
I second Mohammad's sentiments 1000%. Thank you so very much!
very well explained.
So cool to understand
U are a great ❤️
Wouldve been nice to see this closely after i had to prove/show that this was the case, to answer a question that without him mentioning it diredtly was solved. My argument was pretty much the one in the vid was fun to do so.(But i forgot/got confused about the conditional convergence and thought about that this wouldnt be true because if the terms get bigger than you cant ensure you can actually reach any number that easaly anymore, but from the monton decreasing value nature of the series that is quite obvious, to me now, was also define that way just forgot)
I think there is an issue at min 15: absolute convergence is achieved if and only if positive AND negative parts converge (it would be interesting to insert the demo by the way it is a very direct and easy one). So if series conditionnally converge then positive OR negative part diverges.
Many thanks for sharing ressources, impressive work!
Best
"absolute convergence is achieved if and only if positive AND negative parts converge"
This is correct.
"So if series conditionnally converge then positive OR negative part diverges."
That is incorrect. If a series conditionally converges, both the positive AND negative parts _must_ diverge. Conditionally convergent is not the complement of absolutely convergent. There is a third case: divergent.
If only the positive part diverges, then the whole series diverges to infinity. If only the negative part diverges, then the whole series diverges to negative infinity.
Note that it is possible for a divergent series to have both the positive and negative parts diverge, so "both the positive and negative parts of the series diverge" is necessary for conditionally convergent, but not sufficient.
it's riemann not reimann.
8:49 you write -4 instead of -1/4
Sir. In alternative harmonic series
Can we rearrangement diverged to infinity?
How to rearrangement sir?
Learn some grammar and you can rearrange the terms such that they diverge to positive or negative infinity.
Sorry. What you mean 🤔
The typical strategy to rearrange any conditionally convergent series so you get a sum of M is to keep adding positive terms until you get above M, then keep adding negative terms until you get below M, and repeat this process.
To get it to diverge to infinity, you can't use such a simple process as "keep adding positive terms until you get past infinity" because that will never happen, so you won't include any negative terms at all. Instead, what you can do is something like this:
Keep adding positive terms until you get past 1. Then keep adding negative terms until you drop down to 0.
Then keep adding positive terms until you get past 2. The keep adding negative terms until you drop down to 1.
Then keep adding positive terms until you get past 3. Then keep adding negative terms until you drop down to 2.
etc.
This method will lead to divergence to infinity.
thats great
I think it's important to mention that convergence is likely only relevant because we have physics that shields us from infinity and allows us to have information and identity, things like the speed of light and the Plank length. If you really want to gain a deeper understanding of this kind of math you'll have to explore that kind of physics I think.