- Nov 9, 2022
The function $\left(1-z\right)^{n+1}\sum_{k=1}^\infty k^nz^k$ is a polynomial of degree $n$ w.r.t. $z$, and the sum of its coefficients is $n!$. This turns out to be properties of Eulerian numbers.
- Nov 9, 2022
The image of a circle with radius $r$ and centered at $C\left(-2f,0\right)$ through a thin lens at $x=0$ with focal length $f$ and centered at $O\left(0,0\right)$ is a conic section with the focus being $\left(2f,0\right)$, the directrix being line $x=f$, and the eccentricity being $\frac rf$.