Sum of 1/n^4 (Fourier Series & Parseval's Theorem) 

If it's popular do it again

This is the best gaming channel in youtube hands down.

Pikachu I choose you..... Pikachu 'thunder bolt attack'..... 'The sum goes from 1 to infinity of 1/n⁴' gets a shock and died 'The sum goes from 1 to infinity of 1/n⁴' is solved

I've seen this identity a lot but I've never seen such a nice, elegant proof for it. This video was great, keep it up!

0:09 Pikachu used Stare: *Its super effective*

Let me get those people names for a second. To solve a series I need to use: - series by Fourier: ok, a historical French guy, good one by the way; - theorem by Parseval: another historical French guy, a bit more obscure, not a knight of Round Table, not Wagner's opera character; - integration method by Lu Chen the inverse of Chen Lu: both (fictional Chinese???) characters, I didn't find anything relevant on Math in Wikipedia with these names, in order to memorize some Calculus methods; - praying, may God quotient rule would not show up or probably another (fictional Chinese???) character called Quo Chen Lu should be memorized; - and now Pikachu, a Japanese manga character... Too much people for me, almost a Legion to defeat a series.

This was a question on a real analysis exam I had. I really love how this is derived and used. I got that one right. Four years later this has helped me, thank you so much.

yeah, im pikachu: Pretty I good K at A calculus C H U

The only acceptable way to teach math.

Tears of joy. I learned Parseval's theorem, Fourier series and odd/even functions -- all from this single video. Thank you for existing, you incredibly adorable Pikachu.

you saved me, I needed this proof to write my monography in maths... I'm completely thankful :')

Amazing video! Thanks for helping me prepare for my AP Calculus BC test dad

bprp: *wears costume* audience: *surprised pikachu face*

That's very nice . Continue uploading videos like this . It's awesome

Blackpikachuredpikachu!!!! :3

More videos like this please ! 😂😂😂 I LOVED the costume !!

0:15 That's the second most lovely Pikachu I've ever seen The No.1 is of course 0:08

This channel is just amazing.

Were you just in Dusseldorf? There was Japan day, when you released this video. And you could see there many pikachus!

Idk why but u made me spill my water lmao BTW nice video as always!

Wow...pikachu...😆😆 btw u r looking cute..😊

😂😂😂 the first part. Great video. Saw u first time without spectacles

A wild e^x appeared. Pikachu used differentiate: d/dx (e^x) = e^x. It was not very effective...

Just what I needed to start my morning.

OMG i've never seen before the method you use at 2:58 to integrate by part, it's awesome !

Thank you, very very interesting, and the Pikachu outfit is really cool, we want more! I had one doubt about this proof: why does it work at all, considering that x^2 is not a periodic signal (hypothesis of the Fourier series and of the version of Parseval’s theorem you used)? Plotting the reconstructed signal with a0 and a dozen of an, it becomes clear that the function used in the proof is not x^2 over R: it is the repetition along R of the function x^2 defined over the closed interval -pi, pi, which is of course periodic. Thanks!

Hahah 😂 Oi pikachu, wise choice of doing it with Parseval theorem and y=x^2, the version without the theorem yet with the use of y=x^4 was a nightmare of a chan lu xD

How about a general formula for summation of I/(n^(2k)) as n goes from 1 to infinity. Now that would be a thunderbolt for Pikachu.

I love this Pikachu fan who switches pen with a lightning

Is there a general formula for sum of 1/x^n, for all x element of positive integers?

BPRP Pokemon's theorem: The number(n) of likes in this video is directly proportional of the number of Pikachu occurring in the future, where n is natural number and tends to positive infinity

But I thought Pikachu is a detective, not a maths professor.

Pikachu used Fourier series.... It was super effective!

I wish so, so, so much that this man were my calc1 instructor.

Of course, a^2+b^2=(a+b)^2. Profit.

THIS VIDEO WAS LIFE SAVERRRRR!!!! math methods made easy!

Loved the new looking 😄 good job.

pika pika pikachu...it means i like your work.

Detective Redpen solves the case again.

Next: Zeta(pi) U will need pikachu again ;D

I really like your videos, could please make a video on the concept of locus, i'm pretty much confused about the topic.

3:08 TECHNICAL DIFFICULTIES, PLEASE STAND BY. Error code: HTTP-404: Full Blue Pen Not Found.

And he continues to solve the world's problems...

Hey BPRP. This video got me so interested in if there is a way to calculate f(x) so the part of the sum (an)^2+(bn)^2 is a specific serie i wanna calculate?

I have never seen the fourier series written like that... Shouldn't it be the sum from t = negative infinity to t = infinity of a_n * e^int Where the a_n coefficient is 1/2pi integral of f(x) e^-inx dx

Great video! But you forgot to say that the procedure is only true if we consider the function with period 2π, for any other period the coefficients would change a little

Thankyou so so so much for helping out when I needed this explanation the most.... Just loved the way you explained it....😍🥳

more fourier pls! Really enjoyed the video (also more pikachu pls)

Can you prove Parseval’s theorem

Fantastic ...... But I have got quastion Do you think with solution or you found it when you searsh ?

If you want to use this method to find what a sum converges to in general, how do you pick the function? Do I just chose the square root of denominator like you did here? For example, if I wanted to find the infinite series of 1/n^6, would I choose x^3 for my function?

I wanted a suprised pikakuchu meme reference xD. Ps: nice video as always

Between the math and the costume I feel like I'm having a fever dream.

how to know which function f(x) I should take for a given sum? (for exemple sum (1/(2k -1)^2) as k -> infinity)?

Thank you!

If we want to approximate pi, we multiply both sides of the infinite sum by the denominator and take the nth root.so which sum is more accurate when approximating pi for the same number of terms The sum of n^6,n^4 or n^2?

*Dresses as Pikachu* Holds a Pokeball 😂 love that vid keep it up

OH MY .... LORD! THE MATHS!!!

blackpenredpen: sin n pi Me: senpai!

OH MY . . . Fourier and Parseval . . . seriously? Have you EVER introduced them in your previous videos yet?

For the sum of reciprocals of sixth powers, let's just call it S let f(x)=x^3 a_0=0 because x^3 is odd a_n=0 because (x^3)(cos(nπ)) is odd Using DI method, we can find out that, b_n =cos(nπ)(-2π^2/n + 12/n^3) =(-1)^n × (-2π^2/n + 12/n^3) so using Parseval Theorem and substituting the sum of reciprocals of fourth powers (π^4/90) and the sum of reciprocals of squares (π^2/6), we can find out that, 2π^6/7 = 144S - 8π^6/15 + 2π^6/15 144S = 16π^6/105 S=π^6/945 so the sum of reciprocals of sixth powers is π^6/945.

Mr. Cao, if f(x)=sqrt(x), what does this mean for the Riemann zeta function of 3? Does this mean that the Riemann zeta function of 3 cannot be written in closed form?

This infinite series converges to an irrational numerical quantity. For even powers of the series, the sum contains pi raised to the same even power. Is this the case with odd powers of the series? I salute the French Mathematicians.

Does it matter if the problem specifies a different interval? Like in the video, integrals are from -pi to pi, but if the problem states x^2 is from 0 to pi, then should we integrate on that interval instead?

I choose you, x^2!

Why the Pikachu costume???

how can i deal with sigma 1/(2k-1)^4? k from 1 to infinity

What about Onix next? Or maybe Snorlax

I have a question!How can you use Fourier series for a non-periodic function??

this makes me wonder. If you or Peyam got to play PMD, would either of you get Pikachu as the Pokemon you become?

hey BPRP . what is the function for the exercise ? still x² ???

Which university do you teach at?

I've never seen Parseval's theorem before though. Is this something you prove on your course? I would also question how valid the proof is: given that Fourier series are 2pi-periodic, clearly the series can only converge on an interval of length 2pi. Is this sufficient to be able to use Parseval?

the more i look at the thumbnail, the more that pikachu creeps me out

Awesome bro. Very well explained 👏

Is it possible to find a formula to work out the sum from 1 to infinity of 1/n^(2m)?

how can i do with (-1)^k/2k+1 and f:x? thanks

Can you please do a video on the cauchy condensation test for series?

This only works for sums of even powers of the denominator as square roots don't work for negative numbers UNLESS you want the sum of real numbers to give you a complex number

When RU-vid pushes Pikachu content because of a movie but you have a math channel to run.

Integral 1/(x^2-1)^2 dx please

how do i explain to my friends that a pikachu is teaching me calculus?

holy crap pikachu does actually talk

hey bro can you please integrate 4x/(1+x)(1+x^2)^2

My life is officially complete and everyone who watches this is immortal.

This is how we learned it in calculus

What if i want to choose n is a trigonometry function, will it work?

Coud you explain another proof of this formula which involves decomposition of sin(x) into infinite product?

_PIKA PIKA!!!_

Osea que con este método se puede hallar la sumatoria de cualquier serie ?

Why does Pikachu sound so unusual? I expected a bunch of pika pikas

integration of x^3/e^x-1..... Plz solve it

Well done pikachu,

Okay, but how will Bulbasaur solve this?

Pikachu used *calculus skills* Summation has been defeated

Can you prove it without using parseval's formula???

blackpen redpen greenpen yellowcostume

It's been too long since I watched these when I wasn't studying😂

pi^6/945 i think. I might have made some mistakes.

Yes. Try to make sum of 1/n^3 and 1/n^5 also...