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Let $x_{1},x_{2},\cdots,x_{n}>0$,and $x_{1}+x_{2}+\cdots+x_{n}=n$,show that $$\sum_{i=1}^{n}\dfrac{i}{1+x_{i}+x^2_{i}+\cdots+x^{i-1}_{i}}\le\sum_{i=1}^{n}\dfrac{i+1}{1+x_{i}+x^2_{i}+\cdots+x^i_{i}}\tag{1}$$

I try prove following $$\dfrac{i}{1+x_{i}+x^2_{i}+\cdots+x^{i-1}_{i}}\le\dfrac{i+1}{1+x_{i}+x^2_{i}+\cdots+x^i_{i}}$$ $$\Longrightarrow \dfrac{i(1+x_{i}+x^2_{i}+\cdots+x^i_{i})-(i+1)(1+x_{i}+\cdots+x^{i-1}_{i})}{(1+x_{i}+x^2_{i}+\cdots+x^{i-1}_{i})(1+x_{i}+x^2_{i}+\cdots+x^i_{i})}\le 0$$ or $$\dfrac{ix^i_{i}-(1+x_{i}+\cdots+x^{i-1}_{i})}{(1+x_{i}+x^2_{i}+\cdots+x^{i-1}_{i})(1+x_{i}+x^2_{i}+\cdots+x^i_{i})}\le 0$$ it seem this numerator not alway hold,$ix^i_{i}-(1+x_{i}+\cdots+x^{i-1}_{i})\le 0$? so this inequality $(1)$ How to prove it? maybe use induction to prove? Thanks

math110
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  • There's a category error in this question, $x_1 + \cdots + x_n = n$ makes no sense, you're mixing indices and variable values. What actually is $n$? – Thomas Apr 19 '20 at 13:49
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    @Thomas Why does $x_1+x_2=2$ make no sense? – user Apr 19 '20 at 19:28

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