Question

O numero real x que satisfaz a sentença: 1/$1/ x^{1} + 2/ x^{2}+4/ x^{3} ...=1$$1/ x^{1} + 2/ x^{2}+4/ x^{3} ...=1$$1/ x^{1} + 2/ x^{2}+4/ x^{3} ...=1$

Answer 1

       $\dfrac{1}{x}+\dfrac{2}{x^2}+\dfrac{4}{x^3}+\dfrac{8}{x^4}+\cdots = 1\\ \\ \\ \dfrac{2}{x}+\dfrac{4}{x^2}+\dfrac{8}{x^3}+\dfrac{16}{x^4}+\cdots = 2\\ \\ \\ 1+\dfrac{2}{x}+\dfrac{4}{x^2}+\dfrac{8}{x^3}+\dfrac{16}{x^4}+\cdots = 3\\ \\ \\ \text{Recuerde que: }\dfrac{1}{1-a}=1+a+a^2+a^3+a^4+\cdots\\ \\ \text{donde la suma converge si }|a|\ \textless \ 1\text{. Entonces tenemos}\\ \\ \\ \dfrac{1}{1-\dfrac{2}{x}}=3\\ \\ \\ 1-\dfrac{2}{x}=\dfrac{1}{3}\\ \\ \\ \dfrac{2}{x}=\dfrac{2}{3}\\ \\ \\ \boxed{x=3}$