A very interesting factorial based sum

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A very interesting factorial based sum

Рет қаралды 5,319

Күн бұрын

Full solution development for this cool series problem using a few special functions we've become accustomed to here on the channel.
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Пікірлер: 50

@kingzenoiii Ай бұрын

ah yes, the gamma and error functions in their natural habitat.

@mcalkis5771 Ай бұрын

Man I swear you gotta make a video deriving all those formulas of the Beta function.

@oanceatudor4443 Ай бұрын

This isn’t factorial based, this is just based

@jocabulous Ай бұрын

it isn't just based, it's based!

@Player_is_I Ай бұрын

But it isn't just based, it's based!

@GearsScrewlose Ай бұрын

You brought the error down to earth.

@ralfbodemann1542 Ай бұрын

Dear Kamal, I deeply appreciate the ERFfort you put into the evaluation of this sum!

@ayushrudra8600 Ай бұрын

tbh the hardest part of this problem was completing the square 😭😭

@insane_mind Ай бұрын

not that hard, using normal distribution is easy in this case

@GeraldPreston1 Ай бұрын

@@ayushrudra8600 you think that's hard? try completing the circle 😶

@ayushrudra8600 Ай бұрын

@@GeraldPreston1 lol

@antoine2571 Ай бұрын

if you have ax²+bx+c, then it can be rewritten as a(x+(-b/2a))²+smth else the smth else can be easily figured out but I'm too lazy to do it

@CM63_France Ай бұрын

Hi, "Terribly sorry about that" : 3:02 , 3:38 , 6:01 .

@Player_is_I Ай бұрын

Love it😂

@bilkishchowdhury8318 Ай бұрын

There's nothing we can do

@GeoPeron Ай бұрын

Why did the x in xᵏ decompose? Because it was put "to decay" (to the k).

@maths_505 Ай бұрын

@@GeoPeron 😭😭😭

@umairbutt1355 Ай бұрын

Have this like and get out

@gambitito Ай бұрын

error function 🤮👎👎👎

@gambitito Ай бұрын

why does the thumbnail have index n but video has index k? clickbait?!?!

@stefanalecu9532 Ай бұрын

Yes

@Player_is_I Ай бұрын

Fr!!! Scam 2024

@MyOldHandleWasWorse Ай бұрын

Oh man, I had no idea this was going to turn out like that. Fubini's theorem came in way more naturally than I'd anticipated. Well-done, Kamaal! I'd love to see more stuff like this. Historically, I've told my Calc II students that infinite series are a completely different beast from improper integrals, but have never been able to give great examples beyond the Basel problem. On the flip side, since I haven't worked with exact values of non-power series I am not familiar with many techniques beyond Parseval's theorem for computing actual values. Even that method I avoid using because it's still not my field.

@mcalkis5771 Ай бұрын

I'm sorry, but how is x^k(1-x^k) = (x(1-x))^k ?

@jeffreyrodrigoecheverria2613 Ай бұрын

Notice that also

@AlexandreApril Ай бұрын

It's a mistake in the video. The correct equation is x^k*(1-x)^k (see min 2:47) and it is obviously equal to (x(1-x))^k.

@Samir-zb3xk Ай бұрын

He's terribly sorry about that

@maths_505 Ай бұрын

Indeed I am

@slavinojunepri7648 Ай бұрын

I stick to this channel because the math is so amazing.

@user-yg5zb9gk6f Ай бұрын

finally the erf is coming home

@maths_505 Ай бұрын

If only the English could say something about like that about a mojor trophy 😭

@MrWael1970 Ай бұрын

Very interesting infinite series. Thank you for you innovative solution.

@lukebenfer100 Ай бұрын

sqrt(pisqrt(e)) is quite the mouthful 😂 great video

@MortezaSabzian-db1sl Ай бұрын

First, we convert the series into a function f(x)=Σn!/(2n+1)!x^(2n+1) Our goal is to find the function. We can use a differential equation to do this f'(x)=Σn!/(2n)!x^(2n) f''(x)=Σn!/(2n-1)x^(2n-1) After performing simple calculations, we arrive at the following differential equation 2y''-xy'-y=0 It is a linear ordinary differential equation with non-constant coefficients d/dx(2y'-xy)=0 which can be easily solved 2y'-xy=a I think everyone can solve this differential equation y=e^(x²/4)(a/2*√π*erf(x/2)+b) Well, we can easily determine the unknown coefficients a=2 and b=0 y=f(x)=√π*e^(x²/4)(erf(x/2)) Σn!/(2n+1)!x^(2n+1)=√π*e^(x²/4)(erf(x/2)) as a result Σn!/(2n+1)!=√(π√e)*erf(1/2)