Top new questions this week:

The Intuition behind l'Hopitals Rule

I understand perfectly well how to apply l'Hopital's rule, and how to prove it, but I've never grokked the theorem. Why is it that we should expect that $$\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x \to …

(calculus) (analysis) (limit) (intuition)

asked by nsanger 40 votes answered by Andres Caicedo 23 votes

The milk sharing problem

I found a book with math quizzes. It was my father's when he was young. I encountered a problem with the following quiz. I solved it, but I wonder, is there a faster way to do it? If so, how can I …

(linear-algebra) (algorithms)

asked by Dimitris 27 votes answered by Henning Makholm 24 votes

How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$

Nice Question: let $x\in [0,2\pi]$, show that: $$\sin{\sin{\sin{\sin{x}}}}\le\dfrac{4}{5}\cos{\cos{\cos{\cos{x}}}}?$$ I know this follow famous problem(1995 Russia Mathematical olympiad) …

(inequality)

asked by china math 22 votes

Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$

Using a numerical search on my computer I discovered the following inequality: $$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$ where $\rho$ is …

(calculus) (special-functions) (closed-form) (conjectures) (hypergeometric-function)

asked by HWﾠ 20 votes answered by Raymond Manzoni 12 votes

How find this determinant $\det(\cos^4{(i-j)})_{n\times n}$

Question: define the matrix $A_{k}=(a^k_{ij})_{n\times n}$ and where $a_{ij}=\cos{(i-j)},n\ge 6$ Find the value $$\det(A_{4})=？$$ My try:since $$\det(A_{4})=\begin{vmatrix} …

(linear-algebra) (matrices) (determinant)

asked by math110 19 votes answered by Ewan Delanoy 5 votes

How to prove there exists $c$ such $f(c)f'(c)+f''(c)=0$

Nice Question: let $f(x)$ have two derivative on $[0,1]$,and such $$f(0)=2,f'(0)=-2,f(1)=1$$ show that: there exist $c\in(0,1)$,such $$f(c)f'(c)+f''(c)=0$$ my try: since …

(analysis)

asked by nanchangjian 18 votes answered by Mark Wildon 7 votes

Square root approximation

Is there any trick to evaluate this or this is an approximation, I mean I am not allowed to use calculator. $$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7...}}}}}$$

(limit) (radicals)

asked by user2378 17 votes answered by lab bhattacharjee 38 votes

Greatest hits from previous weeks:

How to study math to really understand it and have a healthy lifestyle with free time?

Here's my problem. I'm studying math and when I really work hard, I think I understand things very good, but that comes at a big cost: in the last few years, I've had practically zero physical …

(soft-question) (advice)

asked by Leon Lampret 229 votes answered by Yuval Filmus 77 votes

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?

(logic) (soft-question) (philosophy) (axioms)

asked by Gabriel 39 votes answered by Qiaochu Yuan 55 votes

Can you answer these?

A question on the minimums

For any given $a_{i},\ a_{i}',\ c\in\mathbb{R},a_{i}\le\ a_{i}',\ i=1,\ \cdots,\ 4$, let $$ S:=\left\{ \left(x_{1},x_{2},x_{3},x_{4}\right)\in\mathbb{R}^{4}:\ x_{i}\in[a_{i},a_{i}'],\ i=1,\ \cdots,\ …

(calculus) (real-analysis) (linear-algebra) (geometry) (inequality)

asked by Gina 3 votes

Reference Request on Order Theory topics

I am looking for some references (especially a good recent book) that covers important topics involving partial orders such as: order polytopes, sorting/selection in partially ordered sets, upper and …

(reference-request) (computer-science) (order-theory) (sorting) (polytopes)

asked by Nizbel99 2 votes

Applications of TQFTs beyond physics

I'm giving a talk at a postgrad seminar on the topic of topological quantum field theories (TQFTs) with a mixed audience of pure and applied mathematicians. As such, I'd like to be able to offer some …

(reference-request) (category-theory) (manifolds) (applications) (quantum-field-theory)

asked by Daniel Rust 6 votes