Euler's Original Proof Of Basel Problem: Σ(1/n²)=π²/6 - BEST Explanation
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- Опубликовано: 29 дек 2023
- This video covers Leonhard Euler's original solution to the infamous Basel Problem! - This is also a re-upload since my previous version of the solution didn't adequately explain a certain crucial step, so I decided to remake the whole thing with a better explanation. I hope you all enjoy, and I greatly appreciate all of y'all's support! :)
Hello everybody! I unlisted my previous video on this from two days ago since I realized it didn't adequately explain a certain important step....and it bothered me, so I decided to remake the whole thing with a better explanation. I really hope you all enjoy, and I greatly appreciate all of y'all's support! Thank you! 🙏 😄
Thank you for remaking the video to better clarify the steps, it greatly does help people understand
@@thewolfmanhulk2927 of course, I never want to leave anyone feeling confused during a video…I wouldn’t like that if it were me. Plus, it’s the least I can do for you guys, truly appreciate all my subscribers and viewers!
A much better proof. Euler was a genius.
@@johnhutson3917 thanks so much for watching, glad you enjoyed it! And yes, Euler was one of the greatest! 👑
@@Mathority1729didn’t rly get what u did in 7:54 from this onward i was confused as hell prolly it ain’t your fault cuz many simpler concepts are hard for me like e as a compound interest or why chain rule is valid and other more
Euler was Indeed an Genius and an Intelligent!
💯! Thanks for watching! Really appreciate it 😄
At 7:34 amongst all the sines and pi stuff, you introduced a tangent. Excellent!
i love all your little tangents haha. thanks for the explanation, was pretty easy & engaging to follow
I’m super happy to hear that! Thanks a ton for watching, I really appreciate you!
hey just came here to say that ive watched all the videos uve posted till now and they are very helpful and interesting! really love this kind of video especially where we get to see how problems have been solved for the first time (historically speaking), it really shows that there are plenty creative ways to approach a problem, i think thats important because in school we always learn to go from A to B without searching for different ways of solving the problem. nice work!
First and foremost, it makes me very happy to know that there’s people like you who find value in my content and enjoy my videos, it’s a blessing! Thank you so much for the support!
And absolutely, one of my favorite things about mathematics is all the different ways to solve a problem. There’s a certain beauty to that! And it’s cool to know the history of it as well!
And certainly, schools often don’t do a good job of getting people interested in mathematics, it’s borderline a sin that they don’t expose people to beautiful results and proofs such as this, which can truly excite the mind. I hope to fill that gap with this channel!
Great explanation 👌
You break the problem down beautifully. Love it!
I’m glad to hear that! Thank you so much for the kind words! 😄
I for the first time saw your channel It was wonderful and very clear explanation. Please keep on making videos.. Thanks for your efforts🙏🙏🙏🙏
Really like your presentation --- Thank you so much!
Wow, that was a cool video! Euler was a super genius
Thank you so much for this explanation, finally got it 😊
Great video dude
Perfection!! U r just brilliant
Thank you so much! I really appreciate it!
@@Mathority1729 i lately didn't use expansion of trig. Functions and this problem was gem of one so yeah, besides happy newyear
@@akultechz2342 I’m super glad to hear that, my goal is to excite people with mathematics, and there’s so many cool things that school doesn’t teach! Happy that you found value in this! And of course, I wish you a Happy New Year as well, and all the best in everything you do in the coming year! 😄
@@Mathority1729 keep forcing us to boggle our brains with these problems because maths is all about fun
Great explanation! Happy New Year, dear math lovers! ❤
I appreciate that!! Thanks for watching! And Happy New Year to you too! 🎉🎆
This video is great!!! One thing though, when you took the limit of sinx/x, you cant actually use lhopital’s rule for that, since that would be circular reasoning
Yep you’re completely right! Somebody else mentioned that as well, and I realized I choked on that part 🤣
Thanks for pointing it out as well, comments like these help me improve the channel! And thanks so much for watching!! 😄
Incredible content. For someone whose primary math source is RUclips it is sometimes hard to find a video that I can understand this clearly without having to check others explanations.
I also watched the gamma function video and it was "diáfano" as we could say in Spanish.
I wonder if you'll treat the transcendence of Euler's number someday
Tk u bro.i found my answer in the vid ❤❤
No problem, my pleasure! Glad I was able to help! Thanks so much for watching! 😄
I've always liked this derivation the most. It's the most elegant, I guess.
It’s my favorite solution for the Basel problem because it takes me back in history to the very steps Euler himself went through when he stumbled across the problem! Today, there’s so many different solutions to the Basel problem, some of which I’ve read through, but this one remains my favorite :)
Thanks again for watching, you’ve been here from my very first video, and it really means a lot 😄
@@Mathority1729 I just got it in my recommended lol. Just luck I guess. Good luck man! You're gonna need it in today's RUclips climate.
Awesome video! I did have one question: when we solve for the coefficient k, we get 1/(an infinite product of numbers all with absolute value >1). Intuitively, this seems like it should evaluate to 0, but that obviously isn't the case given how we use it afterwards. I'm wondering if you had any extra insight on this.
Thanks so much, man!
And that’s a great question, glad you asked! Think of it this way, the constant alone, in isolation, would indeed go to zero. However, the constant is not in isolation. Instead, it is coupled with the infinite polynomial, and is distributed across the terms/linear factors of the polynomial. And we don’t want to lose the precision of the constant by essentially evaluating it before coupling it with the context of the polynomial it belongs to.
The constant, In isolation, it is:
1/[(-π²)(-4π²)(-9π²)…]
But multiplied with its polynomial P(x)=sin(x)/x, we have:
[(x²-π²)/(-π²)]•
[(x²-4π²)/(-4π²)]•
[(x²-9π²)/(-9π²)]•…
And evaluating P(x=π/2) for example, we get:
sin(π/2)/(π/2) = 2/π
= [((π/2)²-π²)/(-π²)]•
[((π/2)²-4π²)/(-4π²)]•
[((π/2)²-9π²)/(-9π²)]•…
= (3/4)(15/16)(35/36)(63/64)(99/100)…
This example above is actually the famous Wallis Product for 2/π, and it’s only possible thanks to the constant pitching in (all those terms in the denominators)!
This is a great example to show how when the constant is included back into the function it belongs to, the final product converges when evaluated at real values of x.
I hope this helps explain your question!
Thanks so much for watching!
@@Mathority1729 Thanks for the explanation!
What software do you use for making these vids?
I am having trouble making intuitive sense of this from 4:40 onwards.
question 1 : when x->0, (sin x)/x evaluates to 1. And on the right side you are putting x as zero. wouldn't that mean dividing the left hand side by 0 which is not allowed? maybe I am confusing the basic calculus here.
question 2 : after this step, you are arriving at a value of k. But this is a value of k evaluated at x->0. How can we use this value of k in the main formula and say that is the value of k for "all values of x"?
lim at zero of a function that can be evaluated at 0 is just the value of that function at zero. k is a constant so if k at zero is a value, its always that value
Never heard of the 1644 thing before, super interesting
Yeah it’s insane, what a coincidence!
Thanks for watching the video, I really appreciate it! 😄
@@Mathority1729 No I appreciate you making an amazing video like this in the first place! It’s not easy
@@Daniel-yc2ur that means a lot to me! It definitely isn’t easy haha. Frequent failed takes and edits to get it nice and cleaned up before posting. Takes planning and time, but I’m glad people enjoy them 😄
@@Mathority1729 keep it up man
When finding value of "k" you used limit at 0.Is the value of "k" then satisfy every "x" meaning every point in real number line?
My thoughts might be wrong. But can you elaborate this?
great video!, the only problem i see is that you can't prove lim sin(x)/x by l'hopital, other than that amazing work
Thanks so much, i really appreciate it! And yeah, I should have said smn like “you can verify this using l’hopitals rule” instead of “you can prove this”… lol. But noted for the next time. Thanks for pointing that out!
Doesn't the squeeze theorem explain the limit?
@@vishtoxic5928 yep :)
And so on and so forth...
How can you divide both sides of x if x may equal 0?
@@Larbitoso_o you’re correct, but only when he computes the limit of sin(x)/x. That had nothing to do with his next step of dividing both sides by x
3:22 A circular reasoning! How do we know that sin differentiates to cos?
Draw a simple right angle triangle. U ll get it.
@@sujitmohanty1try to work out the derivative of sin using the limit definition
For the sum of the even terms, couldn't you just state that it is a quarter of all the terms?
For every even term (n), it is exactly a quarter of 1/(n/2)^2, and every even term maps 1:1 to a term of half n.
Regarding the cosine method, (without having done it) I feel that it would spit out the sum of the odd terms, given the magnitude of the first roots are half the difference between adjacent roots.
You’re absolutely right! The sum of the even terms is a quarter of the sum of all the terms for that exact reason! Thanks for sharing the insightful comment for all to see, this is a fantastic way to think of it!