For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

learn more… | top users | synonyms

36
votes
2answers
982 views
+50

Prove that $\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots.$

Question. Let $$ f(x)=\!\left\{\,\,\, \begin{array}{ccc} \displaystyle{\left\lfloor{1\over x}\right\rfloor}^{-1}_{\hphantom{|_|}}&\text{if} & 0\lt x\le 1, \\ & \\ 0^{\hphantom{|^|}} &\...
6
votes
1answer
67 views
+200

What is the regularity of the greatest eigenvalue of the Hessian matrix?

Let $f:\mathbb{R}^n\to\mathbb{R}$ be twice continuously differentiable, i.e. $f\in\mathcal{C}^2\left(\mathbb{R}^n\right)$. Define $\lambda_f:\mathbb{R}^n\to\mathbb{R}$ as the function which associates ...