6.11 Solve the equations:

a) $2 x+1+x^{2}-x^{3}+x^{4}-x^{5}+\ldots=\frac{13}{6}$, where $|x|<1$;

b) $\frac{1}{x}+x+x^{2}+\ldots+x^{n}+\ldots=\frac{7}{2}$, where $|x|<1$.
6.11 Let's use formula (6.10), valid under the condition $|x|<1$:

a) $2 x+1+\frac{x^{2}}{1+x}=\frac{13}{6} \Rightarrow x_{1}=\frac{1}{2}, x_{2}=-\frac{7}{9}$;

b) $\frac{1}{x}+\frac{x}{1-x}=\frac{7}{2} \Rightarrow x_{1}=\frac{1}{3}, x_{2}=\frac{2}{3}$.

Answer: a) $x_{1}=\frac{1}{2}, x_{2}=-\frac{7}{9} ;$ b) $x_{1}=\frac{1}{3}, x_{2}=\frac{2}{3}$.