Partial Sums: The partial sums for this equation are 1 and 0. These are also solutions to the infinite series. I understand that the series never approaches 1/2. I also understand that this is a divergent series and that you can not find the limit/sum of it. Yet, this still has applications in science and is generally accepted as a way of computing the theoretical sum. Thanks for watching! Also let me know what videos you'd like to see next.
Take a look at Matholoiger's old video named "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co." and he goes the one you did fairly early on in the video.
This remembers me about people talking about infinite plus 1 is bigger than infinite. This is very confusing. Im definitely a more confused person than 10 min ago
@@novajourney because it kinda looks weird when I think it with space, like an integer 1,000... Have infinite zeros if I write all those it would need infinite space but infinite plus 1 is bigger and it bugs me because if write this other zero where would it be? I know that math doesn't need to make sense in this way tho
Well, part of the problem is that you're thinking of something with infinite zeros as an integer, which isn't possible even in this realm of math. Infinity isn't an integer, or even really a number, but more of a concept. If I have an infinite amount of something, then if I were to add more to that something, then I would still have an infinite amount, in that sense, the two values would be equal. Does that answer your question?
@@novajourney the thing is that I'm inclined more to the "infinite plus one is bigger" team, after watching this video I guess I got a bit more inclined to the the other side and I got a bit bugged because of it
@@novajourney Trying to find the "limit" of a divergent sum like this is equivalent to studying the anatomy of a unicorn. Like sure if we pretend unicorns exist we could probably hypothesize about its muscle groups; but unicorns don't exist, and the study of unicorns shouldn't be taken seriously. We have a very precise notion of what it means for a series to converge and this is just an abuse of notation.
I'm not trying to find the limit, I'm trying to find a theoretical sum. And I understand that this isn't practical in everyday mathematics, but it has applications in science. Here's a quote from Wikipedia about the Ramanujan summation: "where the left-hand side [the theoretical sum we found] has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory." In this sense, 'unicorns' exist for the purposes of science. You might not find a real unicorn out in the wild and be able to dissect it to learn about it's anatomy. But by creating theories about a unicorn's anatomy, it can have practical applications for learning the anatomies of real animals.
@@novajourney Supposing that you can assign arbitrary values to divergent series, this style of argument leads to absurd conclusions such that 1=0. Paradoxical systems should not be considered really, even if they at face value can arrive at true conclusions e.g. zeta(-1) = -1/12. Math is almost never about the conclusion, rather than the formalization of the process, i.e. proofs
@@CurryHoward777 I understand your argument and I agree with it. I'm also not trying to develop a proof. In the end, I'm trying to make a short video explaining some fairly difficult mathematics to a wide audience and inspire them to learn more. I appreciate your argument, but I can't really respond to this because you seem to be more well versed in this than I am. If you wouldn't mind, could you share some further information about this?
@@ChristofApolinario That's an interesting way of looking at it, but I believe you might have made one mistake when you tried to make this series. In your method of calculation: √2 = 2 ÷ √2 √2 = (1)÷(1)÷(1... => √2 = 1 √2 = 2÷(2÷2)÷(2÷2)÷... => √2 = 2÷(1)÷(1)÷... => √2 = 2 You're getting rid of the parentheses that made the series true, which you can't do. I hope this isn't too daunting. You had an excellent idea, but I wanted to make sure you understood these concepts fully. I hope you learned something and thanks for watching! TLDR: Your series is incorrect and there is some interesting reasons why.
@@novajourney Good explanation but I do have 2 corrections in one of my comments/replies and 1 mistake in your explanation. My corrections: 1. It supposedly 2÷(2÷(2÷(2÷(2÷(2÷(2÷(... Not 2÷2÷2÷2÷2÷2÷2÷...
Partial Sums: The partial sums for this equation are 1 and 0. These are also solutions to the infinite series. I understand that the series never approaches 1/2. I also understand that this is a divergent series and that you can not find the limit/sum of it. Yet, this still has applications in science and is generally accepted as a way of computing the theoretical sum. Thanks for watching! Also let me know what videos you'd like to see next.
Not sure why this got recommended to me but nice little video about a quirk of math. Good luck with your channel!
Thanks!
Take a look at Matholoiger's old video named "Ramanujan: Making sense of 1+2+3+... = -1/12 and Co." and he goes the one you did fairly early on in the video.
Yea, I love that concept, I actually made this video so I could make one about that next.
cool vid!
Great video!
Now do the partial sums.
This remembers me about people talking about infinite plus 1 is bigger than infinite. This is very confusing. Im definitely a more confused person than 10 min ago
lol, what's confusing about it? I might be able to explain.
@@novajourney because it kinda looks weird when I think it with space, like an integer 1,000... Have infinite zeros if I write all those it would need infinite space but infinite plus 1 is bigger and it bugs me because if write this other zero where would it be? I know that math doesn't need to make sense in this way tho
Well, part of the problem is that you're thinking of something with infinite zeros as an integer, which isn't possible even in this realm of math. Infinity isn't an integer, or even really a number, but more of a concept. If I have an infinite amount of something, then if I were to add more to that something, then I would still have an infinite amount, in that sense, the two values would be equal. Does that answer your question?
@@novajourney the thing is that I'm inclined more to the "infinite plus one is bigger" team, after watching this video I guess I got a bit more inclined to the the other side and I got a bit bugged because of it
Yeah, that's understandable. Thanks for watching btw!
Bro, that is an invalid answer because the partial sum doesn't approach 1/2 at all.
u can only give it sum "s" if it converges, since it doesnt converge you cannot give it an arbitrary sum so this is wrong
Um, ok. The Ramanujan summation of Grandi's series is typically accepted as an possible answer, but you don't have to agree with it.
@@novajourney Trying to find the "limit" of a divergent sum like this is equivalent to studying the anatomy of a unicorn. Like sure if we pretend unicorns exist we could probably hypothesize about its muscle groups; but unicorns don't exist, and the study of unicorns shouldn't be taken seriously.
We have a very precise notion of what it means for a series to converge and this is just an abuse of notation.
I'm not trying to find the limit, I'm trying to find a theoretical sum. And I understand that this isn't practical in everyday mathematics, but it has applications in science. Here's a quote from Wikipedia about the Ramanujan summation:
"where the left-hand side [the theoretical sum we found] has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory."
In this sense, 'unicorns' exist for the purposes of science. You might not find a real unicorn out in the wild and be able to dissect it to learn about it's anatomy. But by creating theories about a unicorn's anatomy, it can have practical applications for learning the anatomies of real animals.
@@novajourney Supposing that you can assign arbitrary values to divergent series, this style of argument leads to absurd conclusions such that 1=0. Paradoxical systems should not be considered really, even if they at face value can arrive at true conclusions e.g. zeta(-1) = -1/12. Math is almost never about the conclusion, rather than the formalization of the process, i.e. proofs
@@CurryHoward777 I understand your argument and I agree with it. I'm also not trying to develop a proof. In the end, I'm trying to make a short video explaining some fairly difficult mathematics to a wide audience and inspire them to learn more. I appreciate your argument, but I can't really respond to this because you seem to be more well versed in this than I am. If you wouldn't mind, could you share some further information about this?
Also
2÷2÷2÷2÷2÷2÷... = √2
Where did you find the proof of this?
@@novajourney It's just calculations and trial and error.
√2 = 2 ÷ √2
@@ChristofApolinario That's an interesting way of looking at it, but I believe you might have made one mistake when you tried to make this series. In your method of calculation:
√2 = 2 ÷ √2 √2 = (1)÷(1)÷(1... => √2 = 1
√2 = 2÷(2÷2)÷(2÷2)÷... => √2 = 2÷(1)÷(1)÷... => √2 = 2
You're getting rid of the parentheses that made the series true, which you can't do.
I hope this isn't too daunting. You had an excellent idea, but I wanted to make sure you understood these concepts fully. I hope you learned something and thanks for watching!
TLDR: Your series is incorrect and there is some interesting reasons why.
@@novajourney Good explanation but I do have 2 corrections in one of my comments/replies and 1 mistake in your explanation.
My corrections:
1. It supposedly 2÷(2÷(2÷(2÷(2÷(2÷(2÷(...
Not 2÷2÷2÷2÷2÷2÷2÷...
Taking the log of both sides gets you back to grandhi
2^(1 - 1 + 1 - 1 + ...) = 2^(1/2)
2 /( 2 /( 2 /( 2/ (... = sqrt 2