This is SERIESLY NUTS! Deriving the Sum of Reciprocals of the Odd Numbers Squared using INTEGRALS!
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- Опубликовано: 29 июн 2019
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Let us calculate the infinite sum over the reciprocals of all the odd numbers squared today! It's going to evaluate to pi^2/8 and connects the riemann zeta function of two, namely the Basel Problem pi^2/6 and the Dirichlet Eta Function of 2, namely pi^2/12!
Enjoy! =)
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Haters will say: you can just 3/4 of the sum of reciprocal squares
I'm not a hater but I got that as the solution, then came Papa and gave a whole new way to find the same answer. That's one of the reasons I Love this boi!
@@silasrodrigues1446 yeah it was a joke of mine haha
Or just compare this one with 1/k^2 and they have the same nature
man, that sexy pronunciation of integral always gets me
entegewal kuuuun! uwuuuuu
INTEGERAL
integru
Engineers: Integrals are a kind of sum.
Mathematicians: Sums are a kind of integral.
Goal: being straight
Obstacle: *I N T E GÆR A L*
"let's Fubini this shit" applies to everything everywhere everytime
I did it by adding and subtracting one fourth of zeta(2).
When you have theorems and proofs which rely on Riemann's hypothesis.. You know that papa must prove it one day!
I turn off ad block for this channel
This was so ReciproCOOL.
yo i'm definitely using this for my students lol
I believe there's a way to evaluate this by using Parseval's identity if you want to cover more ways to evaluate this
Normal mathematicians: "In this case we can switch the order of integration."
Flammy: "Let's Fubini this shit!" :D
*uses zeta(2) to solve this*
wait that's illegal
That's like using a 44 magnum to defeat a child proof bottle.
Love that reverse integration evaluation trick! Beautiful video 😊
You should try calculating the sum 1/1^2 + 1/5^2 + 1/9^2 + ••• + 1/(4n + 1)^2 + ••• to infinity.
I imagine it's not too much more difficult because you do the same thing but you do partial fractions twice.
I just did it out and got π²/16+C/2. Where C is Catalan's constant
@@jadegrace1312 Catalan's constant is written as G not C but ok
@@DendrocnideMoroides Well, I am defining it as C. Plus I've seen it written as C and K before, in addition to G, so who cares. It's not like I'm doing anything weird like calling it π or something.
1:29 arbitrary constants: are we a joke to you?
Another great video from Papa
Sexy primes is actually way funnier in spanish; cause in spanish prime numbers are translated as "números primos" where "primos" can also be translated as cousins; so sexy primes is usually understood in spanish as "sexy cousins" and cousin primes as "cousin cousins"
See I like maths but I'm nowhere near this level haha. I'm 17 but I have started studying maths independently everyday for fun starting with the basics so hopefully I'll reach this level one day. I doubt anyone actually cares lmao, but I just wanted to say it.
I'm 15 and still in HS, but I've already self-taught basic calculus lol
Calculus opened my mind and introduced me to everything in a higher level, and now I consume maths and physics videos daily :)
Yeah, I get you. I'm 19 and doing a physics degree. I like papa flammy but a lot of the stuff now seems to depend on being a higher level than I am atm. I'm not familiar with "fubini this shit" or the eta function so it makes it harder to enjoy.
I just want to mention that my age is in between both of yours haha.
welcome to this introduction! we expect great things from you.
@@skeletonrowdie1768 What do ya mean? I assume I'm missing a reference.
Wow your ways of solving problems are just awesome
I have a fun challenge for you I did recently: Find an infinite sum for π by using the arc length of a quarter unit circle and its infinite summation definition of an arc length!
I'm disappointed that sexy primes were not involved in the making of this video.
i love this chanel, really much
I was looking for the sum of the reciprocal of the squares of the factorials. Is that a mathematical constant or can it be derived?
Ey pappa, do u have any opinion on baby rudin? I've heard about this bad boi for long time and want to gear up my analysis. Have u tried it?
Im watching black pen red pen, i know how to meth!
Smooth and elegant
I think the only other detail that was missed here was during the rushed partial fractions where you swapped the addition and subtraction sign a second after 5:25. But no matter because the correct implication was made for A=B because you were just doing your thing my boi and would be able to see the small detail if you slowed the process down. Other than that, this was an exciting video to see and I love the use of integrals and geometric series for this!! Can't wait until the next topic you will have.
The change of index @ 10:10, shouldn't the exponent be n-1? I'm a little confused
How do you know (tx)^2 is between -1 and 1
4:17 : I can say as a french that "Voilà" is the best conclusion word ever
Hello Pappa flammy I liked the video very much. Lately my recommendations have been filled with 50 to 100 ways to do... For example 50 ways to cook a steak. Video idea: 50 ways to do an integral?
Keep up the good work, stay flammable.
Alex
One thing I find interesting about this, the sum of 1/n^2 from 1 to inf is pi^2/6 (basel problem). You could split this sum into two sums over the odd and even integers squared, ie. 1/(2n)^2 and 1/(2n+1)^2 . Intuitively you would think that the even and odd integers each contirbute to half of the full sum, ie. pi^2/12 . But from this you can see that the even integers contribute less that the odd integers since if the sum of odd integers give p^2/8 , then the even integers contibute pi^2/6 - pi^2/8 = p^2/24 , which seems wierd. I wonder if this has anything to do with the idea that in some cases you can't arbitrarily split up the terms in a sum into more sums and expect the same result when you recombine thee results after summing each one individually. Or does this imply there's a difference in the number of even and odd integers (or somthing to that effect). Would love to hear your thoughts Flammable maths, and anyone else's thoughts.
Myles Scollon well since it’s 1/1+1/4+1/9, etc, the first and biggest number is an odd reciprocal, and 1/4, 1/16, etc come nowhere near reaching 1. In that way it makes sense. If the sum started with an even number, I’m sure the even-only sum would then be bigger.
You are the man dud
WHO WILL WIN?
Papa at 9:46 || Papa at 10:55
Der beste Kanal auf youtube. Prove me wrong.
Using the infinite sum starting at 0 of the triple integrals from ln(-1) to infinity/negative infinity (which clearly is -1) of pi factorial + abc dadbdc, I have proved the Riemann hypothesis. Sadly it’s too complicated for the average person to comprehend, so I see no reason to write it down.
Thank God my favorite channel SpaceX is back up
Sir what is value of sum of all factorial
Lets do some physics boi
An elliptical plate initially rests on a horizontal surface at position where its major axis, 4 m, is in vertical position and its minor axis, 2 m, is in horizontal position. Determine the angular velocity of the plate after it is released from rest, at position when its major axis is in horizontal position and its minor axis is in vertical position.
10:55 I hate how you exploited n-1 into n+1... And I hate that it doesn't matter XDD
Brehhh. I know how to meth xD
You can just come up with it using cos(x), It's like 10x easier.
Do the same thing euler did for the basel problem but use cosine instead.
How? Can you sketch the procedure?
I hear it's real hot in Germany and France..stay cool!
Can be done by comparing infinite product expansion for sine and infinite sum for sine
and tyhis is the easiest way
BPRP punching air rn
5:04 it's about here where I regret watching these videos on 2x speed
2:36 In to what therms...😂?
10:09 I have a question here. If you say "k+1=n" then "k" should be "n-1". Why in the video you change "k" by "n+1"? That doesn't change all the answer?
No, since he replaces all k by n-1 it stays the same. On top of that the sum goes to infinity, 1+infinity=infinity.
Arturo Ruiz Y yes, but it really doesn’t matter at all. (-1)^(n+1) = (-1)^(n-1). Try to see if you can prove that yourself.
Also Robin Van Gaalen already made the second point I was going to make. The sum is still an infinite sum
entegewal!
Using the mclaurin expansion of cosine you could do it much easier, but your way was way cooler
Would you explain how?
FM: Made one small mistake
Smart Butts: 1:28 !!!
-(3/2)zeta(-1)π²
best part = 5:02 to 5:10
I need your help, please!
I have a series
Sum_{k=0}^{m} (-1)^k mCk (k+n)^-1 = (m+1) (m+n)C(m+1)
C stands for combinations
Hey this is from Book of Proof!
I SUPPOSE U DELIBERATELY MISSPELT 'SERIOUSLY' IN THE TITLE...RIGHT PAPA?
Do u now that you are the only RUclipsr who replies to me??? Thanks Bruh!!!
At which university do you study in Germany ?
@@PapaFlammy69 Wie kamst du dazu dich für Potsdam zu entscheiden, wenn man fragen darf :D ? Oder war die Entscheidung primär danach gerichtet welche Uni bei dir so in der Nähe liegt ?
Cant u make like a 'previously with papa flammy' thing so we can remember which videos u are referencing
Aw i see m8 sorry to waste ur time, have a flammable day :)
Thanks for the meth on my birthday Papa Flammy!
"Something a first grader can do"
Ey flammy what is the sexy flammy music like a melody behind in a lot of previews. Here 0:22
What if we don't pay you before the end of the course?
2:31 >mfw papa tries to be a sexy boi
Very impresive but can you solve 765138*1427384 mentally?
"Let's Fubini this shit..." Naaaa... "Euly maccharoni" are less tasty than "Fubini al forno".
I'm naturally start to develop a german accent
Flammable maths, what was going on with you and #blackpenredpen?
You said the bprp word🙊
lol
Damn, this video made me uncomfortable
@@PapaFlammy69 5:02
this video is absolutely awesome. a small series on dirichlet L functions and dirichlet series would be great. and also, fuck bprp fans who come and say bullshit. Simply ignore them. Papaflammy ist überlegener als bprp(correct me if the sentence is wrong)
Just do eta+zetta
This video, specifically because of the result, is irrational. Ba dum dum.
I love and I don't know why hhhhh hhhhhh,
rip old comments
No need for any of this work in video.
Sigma(from n=1 to inf ) 1/n^2 -(1/2)^2 *sigma(from n=1 to inf) 1/n^2
π^2/6-1/4*π^2/6 = (3/4)*π^2/6
= 3π^2/24
=π^2/8
I want to be a smart ass
*seriously
Do I see a woooosh?
@@mattgsm I see a double woooosh
@@49fa75 did I see a triple woooosh?
@@mattgsm no u :v
Seriesly Xd
P.S.: Math is about finding unexpected connections
I don't understand the reverse engineering step ... Someone please spell it out.
Edit: I do understand Fubini-ing this shit, but I don't understand where the shit came from.