What is the value of $\displaystyle \frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{3}}}}$ ?

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Use your calculator, summing up what you can. You can also try manually doing it:

$$\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{3}}}}$$ $$=\frac{1}{1+\frac{1}{1+\frac{1}{\frac{4}{3}}}}$$ $$=\frac{1}{1+\frac{1}{1+\frac{3}{4}}}$$ $$=\frac{1}{1+\frac{1}{\frac{7}{4}}}$$ $$=\frac{1}{1+\frac{4}{7}}$$ $$=\frac{1}{\frac{11}{7}}$$ $$= \boxed{\frac{7}{11}}$$