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.
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)
