Archive of posts on Nov 9, 2022

• Nov 9, 2022

  A polynomial whose sum of coefficients is a factorial

  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.

  • Categories: math
  • Tags: combinatorics, number sequence, from zhihu
• Nov 9, 2022

  The image of a circular object through a thin lens

  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 $\frac rf$.

  • Categories: physics
  • Tags: geometrical optics, from zhihu