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Let-x-r-r-1-n-be-n-positive-real-numbers-Show-That-x-1-1-x-1-2-x-2-1-x-1-2-x-2-2-x-n-1-x-1-2-x-2-2-x-n-2-lt-n-




Question Number 198311 by York12 last updated on 17/Oct/23
Let {x_r }_(r=1) ^n be n positive real numbers Show That:  (x_1 /(1+x_1 ^2 ))+(x_2 /(1+x_1 ^2 +x_2 ^2 ))+...+(x_n /(1+x_1 ^2 +x_2 ^2 +...+x_n ^2 ))<(√n)
$${Let}\:\left\{{x}_{{r}} \right\}_{{r}=\mathrm{1}} ^{{n}} {be}\:{n}\:{positive}\:{real}\:{numbers}\:{Show}\:{That}: \\ $$$$\frac{{x}_{\mathrm{1}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{{x}_{\mathrm{2}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} }+…+\frac{{x}_{{n}} }{\mathrm{1}+{x}_{\mathrm{1}} ^{\mathrm{2}} +{x}_{\mathrm{2}} ^{\mathrm{2}} +…+{x}_{{n}} ^{\mathrm{2}} }<\sqrt{{n}} \\ $$

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