A course of pure mathematics, G.H.Hardy
P34
chapterⅠ Real Variables
- 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