# Tagged Questions

51 views

### How prove that $\ln2>{\left(\frac{2}{5}\right) }^{\frac{2}{5}}$? [duplicate]

How prove that inequality $\ln2>{\left(\frac{2}{5}\right) }^{\frac{2}{5}}$?
56 views

79 views

152 views

### How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand?

Is there any way we can prove this definite integral inequality by hand: $$\int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4$$ I don't where to start even, please help. That ...
60 views

### Prove Minkowski's inequality directly in finite dimensions

The typical way I've seen to prove Minkowski's inequality is to take $$|f(x) + g(x)|^p \leq |f(x)||f(x)+g(x)|^{p-1} + |g(x)||f(x)+g(x)|^{p-1},$$ integrate over $x$, and apply Holder's inequality to ...
48 views

131 views

### How to arrange $e^3,3^e,e^{\pi},\pi^e,3^{\pi},\pi^3$ in the increasing order?

For these six numbers, $e^3,3^e,e^{\pi},\pi^e,3^{\pi},\pi^3$, how to arrange them in the increasing order? This problem is taken from the today test: National Higher Education Entrance Examination. ...
### Prove that $\exists k>0$ such that$\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}<n-2014$
Consider a positive sequence $\{a_{n}\}$ such that $a_{n+1}>a_{n}$, and $\{a_n\}$ is unbounded. Show that there exits a positive integer $k$ such that, when $n>k$ ...