数学题 - 1

A course of pure mathematics, G.H.Hardy

P34

chapterⅠ Real Variables

  1. If a,b,c are positive,and \( a+b+c=1 \),then
    \( (\frac{1}{a}-1)(\frac{1}{b}-1)(\frac{1}{c}-1) \geq 8. \) (Math. Trip. 1932)

if a>0 and b>0 then \( \frac{a+b}{2} \geq \sqrt{ab} \)

\( a>0, b>0, c>0, a+b+c=1 \)

\( (\frac{1}{a}-1)(\frac{1}{b}-1)(\frac{1}{c}-1) \)

\(=\frac{b+c}{a} \times \frac{a+c}{b} \times \frac{a+b}{c} \)

\( \geq 2 \times \frac{\sqrt{bc}}{a} \times 2 \times \frac{\sqrt{ac}}{b} \times 2 \times \frac{\sqrt{ab}}{c} = 8 \)

求极限

p207, No.5

\( \lim_{x \to \infty} \sqrt{x}(\sqrt{x+a}-\sqrt{x}) \)

p207, No.8

\( \lim_{x \to \infty} x^3(\sqrt{x^2+\sqrt{x^4+1}}-x \sqrt{2}) \)

求原函数

p281, No.53

\( \int \frac{x-1}{(x+1) \sqrt{x(x^2+x+1)}}dx \)

p263, No.6

\( \int \frac{1}{x \sqrt{3x^2+2x+1}}dx \)

有理数跟自然数等势

\( NXN \sim N \)

笛卡儿坐标系第一象限内的整数格点是直观地能一笔画联起来的,我是用y=-x的平行线簇,从原点出发,呈锯齿状把所有整数格子点串起来,这就表明NXN的元素是可以线性化的,即能存储到一个数组里

具体的双射关系是:

\( f(\langle a, b \rangle)=\frac{(a+b)(a+b+1)}{2}+b \)

反函数 \( g(a)=\langle \frac{1}{2N(N+3)}-a, a-\frac{1}{2N(N+1)} \rangle \) ,其中 \( N=floor(\frac{\sqrt{8a+1}-1}{2}) \)

是把横纵坐标联立后解一个一元二次方程得的这个结果