Question

(P.G.) Resolva a equação x + x^3/2 + x^5/4 ... = 1

Answer 1

$\large\boxed{\begin{array}{l}\rm x+\dfrac{x^3}{2}+\dfrac{x^5}{4}+\dotsc=1\\\\\rm x\cdot\bigg(1+\dfrac{x^2}{2}+\dfrac{x^4}{4}+\dotsc\bigg)=1\\\underline{\rm Soma\,dos\,termos\,da\,PG\,infinita}\\\rm S_n=\dfrac{a_1}{1-q}\\\\\rm a_1=1~q=\dfrac{x^2}{2}\\\\\rm S_n=\dfrac{1}{1-\frac{x^2}{2}}\\\\\rm S_n=\dfrac{1}{\frac{2-x^2}{2}}\\\\\rm S_n=\dfrac{2}{2-x^2}\end{array}}$

$\large\boxed{\begin{array}{l}\rm x\cdot\dfrac{2}{2-x^2}=1\\\\\rm\dfrac{2x}{2-x^2}=1\\\\\rm 2x=2-x^2\\\rm x^2+2x-2=0\\\rm\Delta=4+8=12\\\rm x=\dfrac{-2\pm2\sqrt{3}}{2}\longrightarrow x=-1\pm\sqrt{3}\\\rm S=\{-1+\sqrt{3},-1-\sqrt{3}\}\end{array}}$