WHY are we finding pi HERE?
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Beyond the math and before Squarespace...there was peace among the Chalk and Erasers...was a wondrous era that was. now we're stuck here, in constant battle. Friction. ALWAYS FRICTION. Especially between the Chalk and Chalkboards. Once the best of pals. but that's all over because π had to show up in this problem and absolutely wreck everyone's thinking meat. Like the video and comment "bing bong boom, I'm following arbitrary directions for collective, parasocial fun", thank you.
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@kkanden Жыл бұрын
i would love for the smoothie hoodie to be a recurring hoodie in the hoodieverse of michael penn
@arahatchikkatur1906 Жыл бұрын
A slightly simpler way of getting a geometric series from the start is to multiply both numerator and denominator by e^-x. The denominator becomes 1-(-e^(-x)), which can be expanded into a geometric series. This makes the resulting series+integrals easier to work with as well.
@cheedozer7391 Жыл бұрын
That was my first idea upon seeing the integral, too.
@davidblauyoutube Жыл бұрын
@@cheedozer7391 And mine. 😀
@MrFtriana Жыл бұрын
Me too.
@EtienneSturm1 Жыл бұрын
I agree!
@giacomorapisardi877 Жыл бұрын
Yep, I really did not get why it was better to multiply by e^x - 1 at the beginning...
@QuantumHistorian Жыл бұрын
13:28 Indeed, 1 minus 1/2 is 1 makes all of maths substantially easier.
@ianmichael5768 2 ай бұрын
I replayed that moment multiple times to make sure I heard him correctly. After feeling foolish for replaying it, I found it interesting. Cheers
@alexanderst.7993 Жыл бұрын
"π is that one guy who is never invited to parties,yet still shows up." - a brilliant commenter who's not me
@ChristianRosenhagen 2 ай бұрын
I love how you levitate through the algebraic transformations.
@MrFtriana Жыл бұрын
Ah yes. An integral related to the Fermi-Dirac statistics. Also, is clever what he is doing here. Doing that difference of squares, avoid a (-1)^n term in the series that makes the work more difficult. Also, we can avoid the D-I method, considering the integral of x e^(-ax) as a Laplace trasform.
@enpeacemusic192 Жыл бұрын
I love the more subtle humor of Michael Penn and the more chaotic humor of our dear editor, its really charming and another reason why i love this channel so much :) (beyond the hard math of course)
@titan1235813 Жыл бұрын
@ 8:50, Blackpen/Redpen... yaaaaaayyyy!!!
@maths_505 Жыл бұрын
Oh this is an example of an absolutely gorgeous integration result that connects the gamma and zeta functions!...I solved that integral on my channel and it was marvelous!!!
@suvosengupta4657 Жыл бұрын
thats why it seemed familiar
@parkershaw8529 Жыл бұрын
Man, I am still waiting to hear why pi is here???
@5alpha23 Жыл бұрын
HAHAHAHAHA, I laughed really hard at the merch monologue at the beginning XXXXDDD that presentation was just superb!!
@igorkuivjogifernandes3012 Жыл бұрын
I loved this one! Your channel is amazing. Keep rocking!
@goose_clues Жыл бұрын
nonono, we need *WHY* there's a pi, not *HOW* we got here.
@Thomas154321 Жыл бұрын
A bit disappointed by the clickbait title. You showed what the answer was, but there was little discussion about why. The title implies some insight that was not there.
@attyfarbuckle Жыл бұрын
He mentions Basel at 13:39 & bing, bong, boom Pi appears
@Daniel-yc2ur 7 ай бұрын
Womp womp
@ianweckhorst3200 6 ай бұрын
Plus the e^x-e^2x is basically like inviting it straight in, pi loves when you compare two exponential functions, especially in an integral
@jackthisout9480 Жыл бұрын
I found a pie on the kitchen counter and I know where that came from. Your pi came out of nowhere.
@GicaKontraglobalismului Жыл бұрын
That is an integral which occurs in the study of a degenerate gas of fermions!
@cheedozer7391 Жыл бұрын
Love your videos Professor Penn! I'm sure I'll love this one, but I'll watch to make sure.
@BikeArea Жыл бұрын
That's s good place to start. 😄
@marcellomarianetti1770 Жыл бұрын
at 3:00 it's trivial that e^-2x is between 0 and 1 for x > 0, because since 0 < e^-x < 1 and e^-2x = (e^-x)^2 it follows easily, we all know that if you square a number that is between 0 and 1 you get another number between 0 and 1
@rjabdel Жыл бұрын
I didn’t catch the answer to the clickbait question! Why IS pi here?
@aronbucca6777 Жыл бұрын
This is what I call top quality content
@andreastoumasis7496 Жыл бұрын
that was nice! i didn't know that integral so no spoilers here. what a great idea to use a geometric series that converges to use dominated convergence theorem, how instructive, and again wow!
@michaelbaum6796 Жыл бұрын
Great subtle solution- thanks a lot Michael👍
@ismaelcastillo188 10 ай бұрын
I've encountered this kind of integrals while dealing with fermi dirac distributions in statistical physics. Really nice stuff!
@gandalfthefool2410 Жыл бұрын
As an engineer, I would integrate it numerically from 0 to a very large number assuming the integral converges and then take the first few significant digits as my answer😂
@jesusmariamuruagamarin9016 Жыл бұрын
OMG!!!!. Que fácil parece todo cuando se ha hecho un montón de trabajo duro. Felicidades
@pyrotas Жыл бұрын
I use to solve it in a slightly different (but totally equivalent) manner. Firstly, one recognizes that the 1/(1+exp(x)) is (barring a sign) the derivative of ln(1+exp(-x)). Integrating by parts, the boundary term vanishes and one is left with the integral of ln(1+exp(-x)). Taking some license at x=0 (heck, after all I am a poor Physicist!) rewrite this log as a taylor series in exp(-x). Swap summation with integration (each term converges very quickly) one finds a nice power series which upon simple rearrangements is just the one written by Penn.
@Ttarler Жыл бұрын
I usually skip forward through ads, but managed to hear the fallout about LaTeX support in Squarespace. This is the right channel for me.
@marcoostheimer1293 Ай бұрын
That's just fascinating. Thanks man!
@xizar0rg Жыл бұрын
It would be interesting to see how a "rough draft solution" might start. (I assume it's just working with exp(-x) and then noticing a difference of squares would be helpful several steps along.)
@petergregory7199 Жыл бұрын
Michael, you make everything look as easy as the square root of six times the solution to the Basel problem.
@nazarsimchuk7326 Жыл бұрын
It's interesting that If we raise x in the numerator to some power k - 1, we will get integral that equals to (1-2^(1-k))ζ(k) Γ(k) which is nice connection to the Riemann's Zeta and Gamma functions. I even thought to propose you that integral to show that it is connected to Basel problem for k=2, and was very surprised to see that video.
@SuperSilver316 Жыл бұрын
Dirichlet Eta Function for the win
@TheEyalYemini Жыл бұрын
but why are we finding pi here?????
@General12th Жыл бұрын
Because this problem is congruent to the Basel problem. Why do we find pi in the solution to the Basel problem? That's a different question.
@beeble2003 Жыл бұрын
@@General12th You can't have it both ways. If this problem is congruent to the Basel problem, then "Why do we find pi in the solution to the Basel problem?" is exactly the same question as "Why do we find pi in the solution to this problem."
@frankhenigman5117 Жыл бұрын
3blue1brown has a nice video on why pi is in the basel problem
@TheEyalYemini Жыл бұрын
@@frankhenigman5117 yeah just rewatched it. I just wondered whether there is any geometric motivation to this integral.
@karimjemel7405 Жыл бұрын
Hello professor, could you make a video about the dominated convergence theorem? We always admit that everything converges nicely but how would someone prove it rigorously? Some examples would be nice
@kkanden Жыл бұрын
after three semesters of calculus and a separate course in analysis and topology i can tell you that it's just something you feel and wave your hands saying that "it clearly follows from [insert appropriate convergence theorem] that this converges nicely"
@EtienneSturm1 Жыл бұрын
That would be nice
@syketuri8982 Жыл бұрын
Dr. Peyam has a video on it if you’re interested
@anshumanagrawal346 Жыл бұрын
@@kkanden Nah, you can justify it and you should. In a lot of cases Dominated Convergence Theorem works easily
@cameronspalding9792 Жыл бұрын
I would have written x/(1+e^x)= (x*e^-x)/(e^(-x)+1) =(x*e^(-x))/(1+e^(-x)) and then expanded 1/(1+e^(-x)) using the familiar expansion of 1/(1+u) for |u|
@GiornoYoshikage Жыл бұрын
Yup, this path is shorter and more obvious. I did the same
@user-en5vj6vr2u Жыл бұрын
Well the (-1)^n is a pain
@maxvangulik1988 Жыл бұрын
Just the final left in calc 2 for me, so i would’ve divided the top and bottom by e^(-x/2) and turned the integrand into (xe^(-x/2)sech(x/2))/2
@maxvangulik1988 Жыл бұрын
“But then what?”, you ask? *good question*
@thelocalsage Жыл бұрын
this is how i got it before watching, i was surprised when i saw the approach he took
@parameshwarhazra2725 Жыл бұрын
Hey Michael, I managed to get a smoothie spill on my math major hoodie that i bought last week.
@nunjaragi Жыл бұрын
always thankful and helpful
@General12th Жыл бұрын
Hi Dr. Penn! Very cool!
@davidroddini1512 Жыл бұрын
To answer the question in the title, my local Bob Evans restaurant has a sign that says pi fixes everything 😉
@ihatethesensors Жыл бұрын
Wow that was cool. Also cool to give a shout out to Redpenblackpen for the DI method.
@Wielorybkek Жыл бұрын
I haven't seen the smoothie spill, my laptop screen has too many smoothie spills.
@sobertillnoon Жыл бұрын
I appreciate the DIY clothes tip at the beginning.
@alexbush9250 Жыл бұрын
Speaking of Merch: I desperately want a shirt that says "Play the same game" with some appropriate image
@wagsman9999 Жыл бұрын
That was fun to watch!
@annaarkless5822 11 ай бұрын
ive been messing around with dirichlet series recently, and from this you can see quite quickly that this is 1/1^s - 1/2^s + 1/3^s -.. at s=2 by a well known integral form of these series, then also that the terms with even denominator are 1/2^s times zeta(s), then that this series is the zeta function minus twice these terms, and so is half zeta(2) and this also lets you find this same integral when the x is raised to some power s in terms of zeta(s)
@yuseifudo6075 9 ай бұрын
The joke at the beginning got me dead
@LenPopp Жыл бұрын
"Why are we finding pi here?" could be a weekly series. Maybe even daily.
@adamnevraumont4027 Жыл бұрын
There is a fun math game called "find the circle". Whenever there is a Pi in a result, find the circle that generates the Pi
@ZipplyZane Жыл бұрын
It could be, but then the video would need to actually give an answer. Yes, that series gets us to pi/6, but why? You have to go to a 3Blue1Brown video to get the answer. Look for something about the zeta function and pi.
@peterjoeltube Жыл бұрын
I've got to say that it is very frustrating to see a title like that but then the video doesn't actually answer the question. I was expecting to see an explanation that provides an intuition for why.
@kmlhll2656 Жыл бұрын
thank you very much sir, but I want to know why the number Pi appear where there is a log or exponential function ?
@arantheo8607 Жыл бұрын
Clear and clean
@morrocansaharam833 Жыл бұрын
You are an international teacher!
@Reliquancy Жыл бұрын
Gives my thinking meat a pleasant feeling.
@MusicCriticDuh Жыл бұрын
ngl, i would love to have that exclusive "smoothie stain" merch xD
@RussellSubedi Жыл бұрын
but WHY are we finding pi HERE?
@faxhandle9715 Жыл бұрын
Which Calculus level course would this be from? I remember a lot of crazy stuff from back in the day, but this one has me wondering. 😁😁😵💫😵💫
@tahirimathscienceonlinetea4273 Жыл бұрын
Hi,Michael we can also use 1/1+e^-x
@gregsarnecki7581 Жыл бұрын
At 11:25, that's just eta(2), which is just 1/2 zeta(2) and thus (pi^2)/12. Maybe a video on the relationship between eta and zeta functions?
@jamesfortune243 Жыл бұрын
I need to buy some merch soon.
@behnamashjari3003 Жыл бұрын
Michael enjoys doing math like a kid playing with a dear toy! 😂
@beeble2003 Жыл бұрын
I misread that as "deer toy" and was wondering why you were being so specific. 🤣
@behnamashjari3003 Жыл бұрын
@@beeble2003 I said DEAR toy meaning a beloved toy.
@beeble2003 Жыл бұрын
@@behnamashjari3003 Yes, I know. And I said I misread your comment.
@donach9 Жыл бұрын
Funnily enough my normal routine involves various exercises, then doing maths and waiting to lunchtime to eat (12:8 diet). Then I settle down to watch KZfaq, starting with a Michael Penn video... with a smoothie. I've already sent my beautiful spouse the link to the store so maybe after my birthday I can have a smoothie covered math hoodie too
@funnyadamsandlervideos6404 Жыл бұрын
I just had this same question in difeq
@mfahrii 28 күн бұрын
Dear Micheal, nice presentation but i can not find the answer of the question "WHY are we finding pi HERE?".
@maxvangulik1988 Жыл бұрын
I was thinking it would be a hyperbolic function
@Simpuls Жыл бұрын
Maybe a bit out of place to comment here when my problem is deeply rooted in other videos, but I still wanted to ask. Are there any solutions on the internet or from you on the problems you assign in the number theory playlist at the end of every video? I have an exam this year and would like to know if what I'm doing is right. Your problems are way harder than the ones in class, but also more interesting.
@minwithoutintroduction Жыл бұрын
رائع جدا كالعادة. طريقة رائعة ستحل العديد من التكاملات
@axelperezmachado3500 Жыл бұрын
"bing bong boom, I'm following arbitrary directions for collective, parasocial fun", oh wait a second....
@matthewodell9129 Жыл бұрын
At 11:30, he makes the assumption that you can add and subtract extra copies without changing the sum. I forget the name of the theorem, but I know I've seen a video explaining that there are infinite sums where, by messing with the order of the terms, you can make it equal literally any result. Is just adding new terms meaningfully different from pairing terms in different ways, and what's the criteria for knowing when you can and can't do something like that? Does anyone know?
@moutonso Жыл бұрын
The sums must absolutely converge, which is to say, that when you take the absolute value of each term, their sum also converges. In this case it's ok! Look up absolute convergence of series to find out more information.
@n8cantor Жыл бұрын
If a series is absolutely convergent, the order of the terms does not matter and any rearrangement will converge to the same sum. Since these sums are all of positive terms, they are most definitely absolutely convergent. See en.wikipedia.org/wiki/Riemann_series_theorem
@briandennehy6380 Жыл бұрын
Ouch, my head hurts
@stevenp7991 5 күн бұрын
This is great although i don't understand why you bothered with DI method rather than simple integration by parts
@numbers93 Жыл бұрын
"Is this a particularly hard integral? --- No." He's right, but he decided to solve it the hard way anyway xD
@cheedozer7391 Жыл бұрын
You know, I feel like I have seen something like this many, many times before. To anyone more knowledgeable than I: Is there a theory behind these integrals?
@MrFtriana Жыл бұрын
I know them because they appear in statistical mechanics when you work in the Fermi-Dirac statistics (when You work with electrons, for example) or the Bose-Einstein statistics (in this case is with photons, for example), and want to find expected values of physical observables.
@bluelemon243 Жыл бұрын
If you muliply the zeta function and the gamma function you will get this integral
@Galileosays Жыл бұрын
So it is half the Basel summation. But why does this one and Basel have a pi?
@PiTheDecimal 9 ай бұрын
I am everywhere.
@epsilia3611 Жыл бұрын
6:50 I stopped the video ... Now what do I do 😨
@deuce2293 Жыл бұрын
cool
@ayoubabid213 Жыл бұрын
Nice , i solved by calling zeta(2)
@tioulioulatv9332 Жыл бұрын
الله يحفظكم
@user-ox5ml5ee9v Жыл бұрын
You literally solved the integral
@RigoVids Жыл бұрын
Who is writing the descriptions?
@hansulrichkeller6651 Жыл бұрын
Lieber Michael! Immer wieder ein Vergnügen, Deine Videos anzusehen! Vielen Dank!
@abdulwahabmuhammed-lw7qf Жыл бұрын
How about x=lnu and papa Faynman handle the rest.
@giacomomosele2221 Жыл бұрын
Yep, that’s a good place to stop
@dulguunnorjinbat6136 Жыл бұрын
Who is making these descriptions 😂😂😂
@MichaelPennMath Жыл бұрын
I am -Stephanie MP Editor
@exoplanet11 4 ай бұрын
but WHY is pi there?
@funatish Жыл бұрын
bing bong boom, I'm following arbitrary directions for collective, parasocial fun i didn't like the video though, see how much of a rebel am i?
@takemyhand1988 Жыл бұрын
At this point just substitute the value for x and draw graph for the equation and find area by some other method
@Daniel-yc2ur 7 ай бұрын
What other method?
@mspeir Жыл бұрын
I love all the gibberish you speak and how convinced you are that it actually means something! 😊😂
@looney1023 Жыл бұрын
This is cool but the title implies that you'd be giving some sort of understanding as to why there's a pi appearing; i.e. exposing the "hidden circle", so it comes off as misleading.
@Daniel-yc2ur 7 ай бұрын
It’s the same reason the Basel problem has pi appearing, which as he mentioned, he’s covered before. So you can check out those videos if you want an explanation
@user-vl4yi8jy9m Жыл бұрын
Their repeated appearance in unexpected places indicates that the universe is spherical and will one day return to the state from which it began
@TheMayzeChannel Жыл бұрын
obligatory description comment
@beeble2003 Жыл бұрын
Meh. Honestly, feeling clickbaited by the title. [I've deleted the accusation that's rebutted in the reply from the channel editor.]
@MichaelPennMath Жыл бұрын
Michael does all the ad reads. I think he was sick when he recorded this one, that's why it doesn't sound like him. -Stephanie MP Editor
@beeble2003 Жыл бұрын
@@MichaelPennMath Thanks for the clarification. That being the case, I'll delete that part of my original comment.
@orstorzsok6708 Жыл бұрын
because pi is everywhere...
@alanwj Жыл бұрын
You never answered the question in the title of the video.
@DOROnoDORO 2 ай бұрын
it's left as an exercise to the viewer
@DavidFMayerPhD Жыл бұрын
π is everywhere in Mathematics because circles and periods are everywhere.
@deadlinefortheendtribulati4437 Жыл бұрын
Just as there are 66 books in the Bible pi is in the Bible and 9900 times out of pi it's because GOD has a plan for it.
@TimwiTerby 8 ай бұрын
Please stop flashing up those “subscribe!!” banners. They are off-putting, and also insulting to viewers who are already subscribed. I am perfectly capable of subscribing to things I like without being cajoled.
@Daniel-yc2ur 7 ай бұрын
I’m sure you are capable of that, he’s just playing the social media game! Don’t take it personally man
@nataliem4434 Жыл бұрын
bad title if you aren't actually going to explain the pi at all >:(
@agrajyadav2951 Жыл бұрын
pi is god
@alikaperdue Жыл бұрын
I really dislike the inline advertising in your video. I can not pay to get rid of it. I am paying google to watch you without ads. Doesn't google pay you any of that to make a profit without selling out? I know others do that, but I appreciate when you don't. I think it is weird to assume that your audience would be ok to have ads forced upon them. I'm not. I don't think I am alone. To be clear: I will not watch TV. I plug my ears when advertising is blasted when out. I have decided not to be lazy and let others fill my free time with their personal interests. So I will NOT be watching this channel if inline advertising continues. I am just one, but I hope there are others who will not stand for "free" content at the expense of time with random comments from random people who I am uninterested in listening to. I would like you to take a stand. Stop doing it and say so. This would make your audience very loyal. Like I want to me. PS: I am a big fan. I love your show. Please don't take it away.
@Daniel-yc2ur 7 ай бұрын
At the end of the day people like Micheal have to get money from making good quality shows like this or they wouldn’t do it. If the ads bother you then skip through them. If the thought of a 1 minute ad read is too much for you to mentally handle than maybe the internet is not the best place to spend your time 😅
@maxpetrochenko5025 Жыл бұрын
@blackpenredpen wow 08:50
@Ricktards 6 сағат бұрын
do you have any hoodies that say "math minor"??
3Blue1Brown
Рет қаралды 6 МЛН