\(1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{3}}}}=1+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{4}{3}}}}=1+\frac{1}{1+\frac{1}{1+\frac{3}{4}}}=1+\frac{1}{1+\frac{1}{\frac{7}{4}}}=1+\frac{1}{1+\frac{4}{7}}=1+\frac{1}{\frac{11}{7}}=1+\frac{7}{11}=\frac{18}{11}\)