每日一题：2020-08-16

每日一题: 2020-08-16

题目: 设$a_1,a_2,\ldots,a_n$ 是正数, 求证

$$\frac{a_2}{(a_1+a_2)^2}+\frac{a_3}{(a_1+a_2+a_3)^2}+\cdots+\frac{a_n}{(a_1+a_2+\cdots +a_n)^2}\lt \frac{1}{a_1}$$

参考思路

左边$\lt \frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{(a_1+a_2)(a_1+a_2+a_3)}+\cdots+ \frac{a_n}{(a_1+a_2+\cdots a_{n-1})(a_1+a_2+\cdots +a_n)} \\ =(\frac{1}{a_1}-\frac{1}{a_1+a_2})+(\frac{1}{a_1+a_2}-\frac{1}{a_1+a_2+a_3})+\cdots +(\frac{1}{a_1+a_2+\cdots a_{n-1}}-\frac{1}{a_1+a_2+\cdots +a_n})\\ =\frac{1}{a_1}-\frac{1}{a_1+a_2+\cdots a_n}<\frac{1}{a_1}=$ 右边 问题得证.