放缩法

Template:Multiple issues 放缩法是通过舍去或添加一些项来构造不等式的一种方法。Template:Refn

例子

1. 求证$\sqrt{{\log }_{2}3}+\sqrt{{\log }_{3}2}<\sqrt{2}+1$Template:Refn

   $\sqrt{{\log }_{2}3}+\sqrt{{\log }_{3}2}<\sqrt{{\log }_{2}4}+\sqrt{{\log }_{3}3}=\sqrt{2}+1$Template:Refn

2. 已知a,b,c,d为正数，求证$1<\frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}<2$

   $\frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}>\frac{a}{a+b+c+d}+\frac{b}{a+b+c+d}+\frac{c}{a+b+c+d}+\frac{d}{a+b+c+d}=1$

   $\frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}<\frac{a}{a+b}+\frac{b}{a+b}+\frac{c}{c+d}+\frac{d}{c+d}=2$Template:Refn

3. 求证$\sum _{k=1}^n\frac{1}{k^2}<2-\frac{1}{n}$

   $\sum _{k=1}^n\frac{1}{k^2}<1+\sum _{k=2}^n\frac{1}{k(k-1)}=1+\sum _{k=2}^n\frac{1}{k-1}-\frac{1}{k}=2-\frac{1}{n}$Template:Refn Template:Refn

4. 设$n\in N^{+}$，求证$\frac{n(n+1)}{2}<\sum _{k=1}^n\sqrt{k(k+1)}<\frac{(n+1{)}^2}{2}$

   $k=\sqrt{k^2}<\sqrt{k(k+1)}<\sqrt{k^2+k+\frac{1}{4}}=k+\frac{1}{2}$

   $\frac{n(n+1)}{2}=\sum _{k=1}^{n}k<\sum _{k=1}^n\sqrt{k(k+1)}<\sum _{k=1}^n(k+\frac{1}{2})=\frac{n^2+2n}{2}<\frac{(n+1{)}^2}{2}$Template:Refn Template:Refn

5. 设a,b,c为直角三角形的三边,c为斜边，求证：$a^n+b^n<c^n(n>2)$

   $a^n+b^n=a^2{a}^{n-2}+b^2{b}^{n-2}<a^2{c}^{n-2}+b^2{c}^{n-2}=c^n(n>2)$Template:Refn Template:Refn