Euler's infinite pi formula generator

Today we derive them all, the most famous infinite pi formulas: The Leibniz-Madhava formula for pi, John Wallis's infinite product formula, Lord Brouncker's infinite fraction formula, Euler's Basel formula and it's infinitely many cousins. And we do this starting with one of Euler's crazy strokes of genius, his infinite product formula for the sine function.
This video was inspired by Paul Levrie's one-page article Euler's wonderful insight which appeared in the Mathematical Intelligencer in 2012. Stop the video at the right spot and zoom in and have a close look at this article or download it from here link.springer.com/journal/283... Very pretty.
If you are a regular and some of what I talk about in this video looks familiar that's not surprising since we've visited this territory before in Euler's real identity NOT e to the i pi = -1: • Euler's real identity ...
0:00 Intro
1:49 A sine of madness. Euler's ingenious derivation of the product formula for sin x
7:43 Wallis product formula for pi: pi/2 = 2*2*4*4*6*6*.../1*3*3*5*5*...
9:16 Leibniz-Madhava formula for pi: pi/4=1-1/3+1/5-1/7+...
11:50 Brouncker's infinite fraction formula for pi: 4/pi = ...
18:31 Euler's solution to the Basel problem: pi^2/6=1/1^2+1/2^2+1/3^2+...
21:51 More Basel formulas for pi involving pi^4/90=1/1^4+1/2^4+1/3^4+... , etc.
Music (all from the free audio library that KZbin provides to creators):
kzbin.info...
Take me to the Depth (chapter transitions)
Fresh fallen snow
Morning mandolin
English country garden
Enjoy!
Burkard

[@Mathologer]

Not much Mathologer action lately, sorry. Just insanely busy at work at the moment. Like pretty much everywhere else, my uni here in Melbourne has moved to teaching online this semester and that means crazy hours for at least another six weeks in my case. Happy to at least get this video out. Not the one I planned to do (sign of permutations) but since I had some of the animations lying around much more doable than one from scratch. Anyway, hope you are all staying safe.

[@2false637]

No problem!

[@JB-ym4up]

@@2false637 pun of the week award.

[@matthewvicendese1896]

High schools online too. I have been making youtube videos for each for one of my classes. I only I had have made them last year, then this would have been easy!

[@Money4Nothing]

I just wanted to say thanks so much for your videos. I'm an engineer with only a engineer's background in math, so I can only follow a reasonable percentage of what you talk about. But I had always loved math, and your videos keep my interest in math fresh and fun.

[@michaelempeigne3519]

no problem but how do you do these animations ?

[@marcoantonio7648]

4:45 "On the right, what do we have. Oh, well, whatever that is, it should be infinity. HMMM" "On the left, we are dividing by zero. Double HMMMM" DEAD

[@JB-ym4up]

Well given he had infinity on one side its a good thing the other side is dividing by 0. 😄

[@gamespc495]

0/0=1 ?

[@kunalsrivastava3a539]

Well , its limits here . The calculus limits

[@ZeDlinG67]

I had my worst two weeks lately, and this 30ish minutes was the first I was feeling joyful. Thank you!

[@zachariahhanson1792]

It is amazing, maths is like intellectual freedom. I think if we were truly free, we'd just see maths alone. Freedom is associated with anarchy (t-shirt!), which just makes things better. Nerd Power!

[@LittleBishop001]

So do you think the North should be represented by positive reals? (referring to your current Mandelbrot Set profile image)

[@maxwellsequation4887]

Maybe... I am just glad I don't have to meet stupid people and stay undisturbed for Physics and Mathematics

[@ZeDlinG67]

@@LittleBishop001 I just like this rotation better ;)

[@PapaFlammy69]

Great video

[@garvett6660]

Flammable Maths hey Papa

[@Pillow12442]

Great job flammable maths

[@gibraltar5059]

[@a.fleischbender7681]

The Auto Algebra Song™ makes me so happy.

[@blue_blue-1]

Well, I do need this happiness badly, because I understand nothing.

[@contaantiga5397]

Earlier I wondered: "When is mathologer posting his next video?" Couldn't have gotten a better answer :)

[@KilgoreTroutAsf]

I first read you comment as "Euler wondered..."

[@aaronleperspicace1704]

Maths.... is the most beautiful thing ever. So subtle, so abstract and so transcendental... it always existed and will always exist. What we know is a tiny fraction, the most will always be beyond our comprehension.

[@FloydMaxwell]

You remove all the tedium, replacing it with slick graphics, perfect music and your happy personality. Your love of math is the "answer" for us. Thank you!

[@sniprsprimordium5625]

Engineers: Hold my Taylor Series

[@henryhayton8784]

No need, sin(x) = x

[@cerwe8861]

@@henryhayton8784 cos(θ)=1

[@darkseid856]

π^2 = e^2 = g

[@erik-ic3tp]

This one: kzbin.info/door/7590VTWe6m0kq3gJcgLINg

[@tgwnn]

@@darkseid856 π^2 being close to g is not a coincidence. "g" depends on the unit system, and the meter was chosen (or it was at least one of the definitions) so that a meter-long pendulum had a period of exactly two seconds (the second was chosen a long time ago as 1/86400 of a day). If this is exactly true, that gives L=g/π^2=1 (in SI units). It is 99.3 cm actually with the standard g, of course it changes a bit by the place on Earth. edit: not saying that you don't know this. but pretty sure at least some people in the comments don't. I found this out many years after learning about "π^2=g"

[@DontMockMySmock]

Euler is quickly rising in the ranks of "potentially the smartest human ever lived" in my estimation.

[@alexpotts6520]

Historians estimate that 40% of all mathematical papers written during the years Euler did his maths, were written by Euler.

[@CosmiaNebula]

@@alexpotts6520 I want a citation for that. It's amazing.

[@blizzbee]

Didn't watch video yet... Did Euler prove every math statements he invent? Just asking.

[@pinchus2714]

@@blizzbee No. Euler's conjecture is a variation on Fermat's Last Theorem and was actually proven false in the 20th century and a counterexample was found. However, he didn't claim it was necessarily true, he simply conjectured it...so I'm not sure what you mean by statements.

[@rickstevens1167]

@@pinchus2714 axioms, lemmas, theorems, conjectures.. Duh

[@TheLetterW736]

This is like a grand unified theory for pi formulas. Amazing!

[@rc5989]

The world is always a bit better with a video from the great Mathologer. I am glad you put this together, thank you Mathologer!

[@ahoj7720]

Beautiful presentation. Euler's original book "Introductio in Analysin Infinitorum" is a treasure. It's easily readable, even if written in Latin! (there are translations, of course.) It's exactly the same spirit as your presentation.

[@portr002]

Every time you release a new video it just happens to be exactly what I needed to see! Thank you!

[@duggydo]

Glad you are back! I was concerned after the channel was hijacked. You are a tier above the other math youtubers in my opinion! :)

[@yaaryany]

Beautiful. I usually do not comment on channels, but i had to on this one. Amazing. Never stop making these videos.👏👏👏

[@genuineprofile6400]

Would love to see about Ramanujan's fast converging Pi series. There are plenty of interesting theorems by Ramanujan. Please demonstrate those.

[@Mathologer]

Very much worth doing but really tricky stuff to get right. I am actually regularly feature Ramanujan's fast converging Pi series exams for one of the courses that I teach at uni. So definitely on the radar ...

[@Mathologer]

Very much worth doing but really tricky stuff to get right. I am actually regularly feature Ramanujan's fast converging Pi series exams for one of the courses that I teach at uni. So definitely on the radar ...

[@kimmalyncleaveway2907]

@@Mathologer connection problem? This was posted twice :)

[@Mathologer]

@@kimmalyncleaveway2907 ??

[@kimmalyncleaveway2907]

@@Mathologer I said, the comment you posted was duplicated, and this error may be due to a connection problem, as I have witnessed such problems on youtube before

[@AstroB7]

Always having a great time watching Mathologer !

[@dougr.2398]

I’ve been waiting for a good lesson on this for many years. Thank you!

[@smiley_1000]

Absolutely charming video, it's amazing how products and sums are linked and how one is able to derive the known formulae so effortlessly.

[@Vatsalreso]

I usually never comment on videos but this video is a masterpiece. I never knew this could be conceptualized so easily. You are a legend.

[@ben1996123]

The general formula for the sum of 1/x^(2n) is hidden in here too, and only requires a few extra steps. Start from the chapter 2 formula and take logs and derivatives to get the formula at 10:40: cot(x) = 1/x + 1/(x-pi) + 1/(x+pi) + 1/(x-2pi) + 1/(x+2pi) + ... Move the 1/x term to the left side, and take 2n-1 more derivatives. The k'th derivative of 1/x is (-1)^k * k!/x^(k+1), so when k = 2n-1 this is -(2n-1)!/x^(2n). So we have (d/dx)^(2n-1) (cot(x) - 1/x) = -(2n-1)! * (1/(x-pi)^(2n) + 1/(x+pi)^(2n) + ...) Now divide by -(2n-1)! and take the limit as x -> 0. 1/(2n-1)! * lim x -> 0 [(d/dx)^(2n-1) (1/x - cot(x))] = 1/(-pi)^(2n) + 1/pi^(2n) + 1/(-2pi)^(2n) + 1/(2pi)^(2n) + ... On the right side, all the negatives are squared away, and we end up with 2 copies of each term. So multiply by pi^(2n)/2, and we get this: pi^(2n)/2 * 1/(2n-1)! * lim x -> 0 [(d/dx)^(2n-1) (1/x - cot(x))] = 1 + 1/2^(2n) + 1/3^(2n) + ... = zeta(2n) This formula looks messy, but there's a trick: notice that on the left, we have something of the form 1/k! * k'th derivative of f(x) at x=0. These are just taylor series coefficients! The left side is really just pi^(2n)/2 times the coefficient of x^(2n-1) in the taylor series of 1/x - cot(x). There are a few other things we can do to make the formula easier to read. We can multiply the function by x to make the powers line up nicely (otherwise the 1/k^8 sum will be related to the coefficient of x^7, instead of x^8). This gives: zeta(2n) = pi^(2n)/2 * coefficient of x^(2n) in the taylor series of 1 - x cot(x) The next thing we can do is move the pi^(2n) "inside" the taylor series, by replacing x with pi x. We can also move the factor of 1/2 into the function. Then we get: zeta(2n) = coefficient of x^(2n) in the taylor series of (1 - pi x cot(pi x))/2, or equivalently, (1 - pi x cot(pi x))/2 = sum n=1..inf, zeta(2n)x^(2n) And indeed, if you ask wolframalpha to compute the taylor series of (1 - pi x cot(pi x))/2, you get pi^2/6 x^2 + pi^4/90 x^4 + pi^6/945 x^6 + pi^8/9450 x^8 + ... Finally, comparing this series to the standard taylor series for cot in terms of Bernoulli numbers gives Euler's general formula for zeta(2n)

[@Mathologer]

Got pretty close to including a variation of this in the video on the Bernoulli numbers and the Maclaurin summation formula.

[@WaltherSolis]

Nice insight

[@danielreed540]

Wow! A way to generate a sum series of the Euler pi expressions. I wonder if any complex analysis (or higher dimensions) can allow formulas & solutions for the odd powers of pi? (I noticed the general "super" formula/expression by Euler would result in imaginary #s (i, or (-1)^(1/2)) when trying to achieve the odd powers.)

[@ben1996123]

@@danielreed540 there is a slightly different formula that can be proved in basically the same way, but for the sum of zeta(n) x^n for n>=2. specifically this sum is equal to -x(gamma + polygamma(1-x)) where gamma is the euler mascheroni constant, and polygamma is the log derivative of the gamma function. then the formula says that the derivatives of this function (call it f) satisfy f^(n)(0)/n! = zeta(n). using the reflection formula polygamma(x) + polygamma(1-x) = pi cot(pi x) allows you to recover the formula from my first comment by computing (f(x) + f(-x))/2, but of course replacing x by -x and adding them will wipe out all of the odd terms of the series.

[@Maniclout]

Nice

[@cerwe8861]

Just watched an old Video and now there is a new Video! Da steckt wirklich großartige Arbeit in den Videos, vielen Dank

[@naimulhaq9626]

Breath taking, awe inspiring and spellbinding. Thank you Euler/Mathologer.

[@danieledaroma6293]

Your videos are AMAZING....and the explanation is always very brilliant!!!

[@IshanBanerjee]

A mathologer video 😊😊😊😊😊 I keep waiting for the great videos like it. Just love watching mathologer videos.

[@maurosobreira8695]

I just went half way the video and WOW! Euler was amazing, but you are amazing too, real quality stuff! Thanks!

[@srila6893]

Enjoyed it. Great insights. Good to see. Stay safe Mathologer.

[@mohammadal-turkistany2151]

Thanks for yet another enjoyable math insight.

[@noonesperfect]

smooth, elegant and graceful always.... thanks Mathologer

[@heeraksharma1224]

Please keep this good work going forever!

[@JCOpUntukIndonesia]

Another great video from prof. Burkard. I love it! Thank you and stay safe professor. Note: I'm amazed that even Euler is Mathologer's Patreon Supporter.

[@AlfW]

That's the most enlightening math video I have watched recently.

[@benjaminbrady2385]

Despite the lockdown, the comforts I appreciate in life have really shown themselves to me and I feel grateful for what I have. These maths animations are among that!

[@fackingcopyrights]

Unfortunately im a big Euler fan and i already knew everything in this video, but i love to see it over and over again. There is never enough Euler.

[@antoniussugianto7973]

Thank you sir.. the contents of your videos are truly wonderful... !! Absolutely one of the bests on youtube...! : )

[@MrBuckelgumpen]

"Just tell our brains to shut up." one of the nicest phrases concerning the understanding of math problems.

[@ellatifijalal]

This reminds me of my high school Math teacher, he used to tell us " and this is the art of mathematics"

[@pkmath12345]

Love the pattern in trig to explain this! Worth trying them out for sure!

[@jagatiello6900]

I'm amazed by how useful are the addition (0) and multiplication (1) identities, two of the five members of Euler's identity, btw... Nice video! Thanks a lot for this great channel.

[@gabey.youngblood2613]

Great video! As I'm wrapping up Calc II, this was a really great way to extend what I've seen of infinite series.

[@Mathologer]

Glad you enjoyed it!

[@ishitamukherjee5183]

Who all agree that this is the best mathologer video ever made? Thanks a lot Sir for this wonderful video.

[@alexpotts6520]

The thing about Euler's solution to the Basel problem is that the maths in it isn't actually very hard to follow, it's sixth-form-level stuff really. But the reason the Basel problem stumped the likes of Fermat and Bernoulli was because the solution requires thinking outside the box, and skating on thin ice with regards to mathematical rigour. Euler took a few steps that more experienced mathematicians might have disregarded as nonsense - but he got the right answer, didn't he? It was Euler's creativity that was what really unlocked the problem.

[@samueldeandrade8535]

Exactly. And now, unfortunately, we live at a time when academic authorities try to prevent students to enjoy mathematics like Euler did, even mocking the ones that dare to do that, because of "lack of rigour". Such awful people.

[@turdferguson3400]

Ooh ooh I've been working on polylogarithm functions and Dirichlet series recently and what you have shown is very exciting to me.

[@turdferguson3400]

To get the sum[1/s**2n, s=1 to s=infinity], just take derivatives of the function (x*cot(x)) 2n times, then multiply by -pi**2n*1/2. I was literally working on this the whole past week.

[@Archipelago.]

*Euler* again ! He was really a badass his work is everywhere メ!

[@nanigopalsaha2408]

28:11 Hey! Euler is your Patreon patron! 28:25 And so is Mandelbrot!

[@Mathologer]

Yes, great, isn't it :)

[@52.yusrilihsanadinatanegar79]

Oh god

[@rolfs5854]

This was a really great journey.

[@Micetticat]

Wow the logarithmic trick was amazing!

[@BlaqRaq]

Great presentation. You made it looked easy.

[@paul21353]

Woww...an amazing insight into how Euler's mind worked. I sooo much enjoyed this video. It involves taking notes and a bit of thought but it is worth all the effort in the end.

[@pritamdey5718]

Great video as always!!

[@richardschreier3866]

Another delightful video illuminating the connections between various infinite sums (and an infinite product) and powers of pi by using the product formula for sine. Glorious. In response to your query about what is missing from the Nike argument, two things stand out for me. If you start with the product formula for sine as a postulate, you need to show that a) the product converges and that b) the leading coefficient needs to be one. I expect there is a theorem that allows you to then conclude that this infinite product which has the same zeros and scale factor as the sine function is indeed the sine function. If you are looking for more potential material, I think the Gamma function, its various identities (the reflection formula, the Legendre duplication formula) and its use in evaluating infinite products is pretty cool.

[@ny6u]

GORGEOUS VIDEO 👏🏻👏🏻👏🏻👏🏻👏🏻

[@heeraksharma1224]

The legend is back!

[@georgeanton8567]

Anyone try to work out Sum_i 1/i^6? Using the method in the video I ended up with Sum_i 1/i^6 = sum_{i,j,k} 1/(i^2j^2k^2) - 2\sum_{i

[@mo6949]

Hey @mathologer , if you're doing a future video on quintic equations ; it would be fascinating to see how even the greatest mathematicians fell short on trying to find a formula and their insights- especially euler ( i can't seem to find his attempt on the quintic).and George Jerrard who was reluctant to accept the quintic was unsolvable by radicals etc . I'm sure there also must've been interesting attempts by ferrari and cardano . Tschirnhaus transformations....... (Fingers crossed) this is the content of your next video

[@Cardgames4children]

Wow, the equality between the partial sums and the partial fractions kinds blows my mind.

[@newlaty72]

Damn Euler! The greatest mathematician of all time. True superhero.

[@pardeepgarg2640]

Isn't Newton or Gauss are greatest

[@newlaty72]

@@pardeepgarg2640 you may be right, but I'll stick with Euler

[@pardeepgarg2640]

@@newlaty72 me with Ramanujan :D Everyone has its own Taste :D

[@xyz.ijk.]

The crisis has kept me busier than ever, and so I am even that much *more* grateful that you posted a video so I could take a good mental health break. PS: Although one can find patterns in anything, it's fun that 28:57 is part of the continuing fraction of 7.

[@0113578774]

just wow ... i am a maths teacher to high schoolers but let me tell u ... i feel like i am a happy little kid with his toy when i watch your videos . you are amazing . thank you

[@helicarbr]

Wonderful as always.

[@wyattstevens8574]

Each term of the continued fraction adds exactly one more term of the sum (via Euler's continued fraction formula)

[@bernhardriemann3821]

Simply fantastic

[@maximussu7609]

Very high quality Mathologer

[@noahbertholon1076]

I'm a french high school student and i don't understand a lot, but what a pleasure to listen to !

[@anshshah6775]

It was truly inspired i loved it😍😍

[@Macieks300]

This is great, I wish we would go even deeper into the maths

[@blue_blue-1]

Wheeeeere do you want to go???

[@nafissaatlagh206]

Dis is pure beauty ♥️♥️😍😍 much love math mathologer ♥️♥️♥️♥️

[@sciencelover65]

I am a student of class 11. But I love mathematics. I love to learn and think. But the teachers I know always teaches the things which is important for our exam. But I am not satisfied with that. That's why Internet is the only source where I can learn. I think this channel is the best. Lots of love for you sir for making these kind of videos.

[@abelferquiza1627]

Great! As ever you do...

[@mehdisi9194]

So nice math video. Thank you so much

[@TrimutiusToo]

Watched 28 minutes video, but only thought: "why so short? I wanted more pi formulas!"

[@Mathologer]

Well there is definitely more material for part 2, 3, 4, ... :)

[@chunchen3450]

Euler formula, Taylor series, and of course the mysterious pi! Thanks for the video, very inspiring. once I can not follow the auto formula, I can easily go to meditation mode, thanks for the music.

[@giuseppemalaguti435]

Quando vedo i tuoi video, devo spegnere la TV, e spegnere la radio.... Devo concentrarmi senza un minimo rumore

[@wesleydeng71]

Wonderful!

[@gereralshenx]

Your videos are great review and sometimes brand new stuff. I had seen this at some other point in time in history of mathematics, but my teacher must have skipped a few steps or I was half asleep.

[@Mathologer]

Some of the things I show in this video you won't find anywhere else :)

[@coAdjointTom]

General note: there is a general product equivalent of the taylor series using something called product calculus. Very much worth checking out and useful in statistics as well!

[@terryendicott2939]

"Simply magic." How true.

[@Tifferan3112]

This is so cool!

[@lordlix6483]

Beautiful video :D

[@phasm42]

That infinite product for sin x, just brilliant.

[@GoutamDas-iq8lj]

Great. Both the proves ant the music.

[@adrianoseresi3525]

O wow! I didn’t know Euler was one of your patrons!

[@ljfaag]

I love the animated algebra :)

[@MYPASSIONMYLIFESTYLE]

24:21 I still don't understand how the orange sum is the square of (π^2)/6

[@davidgould9431]

That got me stumped for a bit, too, but I think I have found a way to explain it: The sum we want is every single term (1/i²)(1/j²) for every combination of i and j from 1 to infinity. So, we have (1/1²)(1/1²) + (1/1²)(1/2²) and so on when the first term is (1/1²). This is (1/1²)(1/1² + 1/2² + 1/3² + ...) Then the same for (1/2²): (1/2²)(1/1²) + (1/2²)(1/2²) + (1/2²)(1/3²) + ... giving (1/2²)(1/1² + 1/2² + ...) Grouping those together, we get (1/1² + 1/2²)(1/1² + 1/2² + 1/3² + ...) And so on for every 1/i² in the first term, giving (1/1² + 1/2² + 1/3²...)(1/1² + 1/2² + 1/3² + ...) = (1/1² + 1/2² + 1/3² + ...)² I hope that helped.

[@MYPASSIONMYLIFESTYLE]

@@davidgould9431 thank you👍👍👍. It was helpful.

[@pullingrabbitsouttaahat]

Good. Enchanted.

[@mikikaboom9084]

This is so cool.

[@ridefast0]

It would be fun to see your interpretation of the BBP hexadecimal digit of Pi extraction formulas, which seems (to me) conceptually impossible for an irrational number and also seems to hint at a subtle relationship between Pi and the number 16. Also, the basic BBP formula for Pi seems to generate better than 1 decimal digit of precision per iteration which seems amazingly fast even if each step requires quite a lot of computation. I enjoy your content, thanks.

[@MathemaTeach]

Im glad to come across your channel. You have very informative videos. Good job!!! 👍i know we are now shofting to online classes, i myself as a teacher felt what you feel. I just launched my you tube channel for the purpose of making this online teaching easier. I have been making videos for my classes lately. We have 2 weeks left of school. Good luck to all of us. Stay safe

[@jamesfortune243]

This video was beyond good, with a limit of great as n goes to infinity.

[@Manoj_b]

Wow, ausome vedio ♥️ ,mathaloger, and I want to say one thing that at the power sum vedio is one of my best favorite vedio ever seen I like that vedio lot and I made an general formula for summation((1/n^m),1,n) .and I like all your vedios and what a animation my special thanks to all of your team and you.....!

[@AvatarBowler]

That anarchist A on your math shirt is fire. 😄

[@EddieEntertainment]

great video!

[@peterdriscoll4070]

Wonderful.

[@kruvik]

omg the music you use to introduce the chapters always remind me of That Chapter, so I expect people to die or to disappear... Nice video though!

[@dhoyt902]

Thank you Lord Mathologer. I would love to see how the Pi continued fraction defined by b's(3 , 6, 6, 6 ...) and a's(1^3, 1^3 + 2^3, 1^3 + 2^3 + 3^3, ...) is derived. Its neat because Pi - 3 is that sum of cubes fraction which is also equivalent to an alternating sum involving (n, k). I would also be supremely interested to know how Nilankantha accelerated the 'leibniz' arctan(1) by multiplying by the square root of 3 on one side and dividing by powers of three in the sum. I just read Pi Unleashed but not able to find answers to these questions.

[@Mathologer]

Pi unleashed, great book. I'll keep returning to pi in the future. Lots more amazing stuff to talk about :)

[@rar497]

Why does 1/(2×3×4) − 1/(4×5×6) + 1/(6×7×8) − ··· = (π−3)/4 ? (One of the "beautiful equation" candidates.) That π/4 looks like arctan(1) from Leibniz's formula, and cf. the generalised continued fraction in Brounker's formula! PS: Also, Viète's formula is rather cute: 2/π = cos(π/4) × cos(π/8) × cos(π/16) × cos(π/32) × ··· Thanks for the channel.

[@devrimturker]

John Wallis, also presented values of ratios of continued fractions

[@LifeIsBeautiful-ki9ky]

Go through the Euler Continued Fraction theorems, It gives simple Formulas for Continued Fractions.

[@MrMetrizable]

Love the shirt!