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Question Number 192481 by Tomal last updated on 19/May/23

Answered by aleks041103 last updated on 19/May/23

$$\frac{1}{2}=\frac{1}{1+x}(1+{\left(\frac{1}{1+x}\right)}^2+{\left(\frac{1}{1+x}\right)}^4+...)$$$$if\mid \frac{1}{1+x}\mid <1,i.e.\mid 1+x\mid >1$$$$\frac{1}{2}=\frac{1}{1+x}\frac{1}{1-\frac{1}{{(1+x)}^2}}=\frac{1+x}{{(1+x)}^2-1}$$$$\Rightarrow x^2+2x+1-1=2+2x$$$$x^2=2\Rightarrow x=\pm \sqrt{2}$$$$\mid 1+\sqrt{2}\mid =1+\sqrt{2}>1$$$$\mid 1-\sqrt{2}\mid \approx 0.4142<1$$$$\Rightarrow x=\sqrt{2}$$

Commented by Tomal last updated on 19/May/23

$$Thanks\lambda \underset{―}{\underbrace{}}$$

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