Let $x_1,x_2,x_3,...,x_n$ be positive numbers. Prove that

$$\frac{1}{1+x_1}+\frac{1}{1+x_1+x_2}+\cdots+\frac{1}{1+x_1+x_2+\cdots+x_n} < \left(\displaystyle\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\cdots+\frac{1}{x_n}\right)^{1/2}$$

this is what I can think of, but I don't know how to continue.

$$n=1$$ $$\frac{1}{1+x_{1}} < \frac{1}{\sqrt[ 2]{x_{1}}}$$ $$1+x_{1}>x_{1}$$ $$a)\text{ if } x_{1}>1$$ $$x_{1}>\sqrt[ 2]{x_{1}}$$ $$x+1>\sqrt[ 2]{x_{1}}$$ $$b) \text{ if }0<x_{1}<1$$ $$x_{1}<\sqrt[ 2]{x_{1}}$$ $$0<\sqrt[ 2]{x_{1}}<1$$ $$1+x_{1}>\sqrt[ 2]{x_{1}}$$ $$\therefore\ x_ {1}\text{ positive}$$ $$\frac{1}{x_{1}+1}<\frac{1}{\sqrt[ 2]{x_1}}$$

  • $\begingroup$ Hint: $\dfrac1{x_1+x_2+\dots+x_k}<\dfrac 1{x_k}$. Actually, inducrion is not necessary. $\endgroup$
    – Bernard
    Sep 18, 2021 at 19:18
  • $\begingroup$ In your proof of $n=1$, it seems like you didn't square both sides. $\endgroup$
    – Calvin Lin
    Sep 20, 2021 at 14:54


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