$1a.$ Let $f(x)=x+\frac{x^3}{2}+\frac{x^5}{4}+\frac{x^7}{8}+$cdots$+\frac{x^{2n+1}}{2^n}+$cdots. Find all $x$ satisfying the equation

$f(x)=-1.$

$1b.$ Let $A=\frac{1}{2}+\frac{1}{3*2^3}+\frac{1}{5*2^5}+$cdots$+\frac{1}{(2n+1)*2^{2n+1}}+$cdots. Show that