A very interesting factorial based sum

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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[mcalkis]

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

[kingzenoiii]

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

[ralfbodemann]

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

[oanceatudor]

This isn’t factorial based, this is just based

[CM_France]

Hi,

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

[ayushrudra]

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

[MyOldHandleWasWorse]

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.

[MortezaSabzian-dbsl]

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))

as a result
Σn!/(2n+1)!=√(π√e)*erf(1/2)

[GeoPeron]

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

[gambitito]

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

[MrWael]

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

[slavinojunepri]

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

[mcalkis]

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

[lukebenfer]

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

[jmontgomery]

At 3:16 did I miss something? Isn't the integrand supposed to be x^k*(1-x)^k ... not x^k*(1-x^k)?

[Lolo]

3:30 is that incorrect? Should be (x^k)*(1-x^k) = x^k - x^(2k); not (x-x^2)^k

[GeraldPreston]

sums are just budget integrals
nice vid

[UnknownGhost]

This wasnt the hardest one its all basic properties of integral and gamma functions😊😊