$$2 + \frac { 1 } { 3 + \frac { 1 } { 1 + \frac { 1 } { 1 + \frac { 1 } { 2 } } } }$$
$\frac{41}{18}\approx 2.277777778$
$$2+\frac{1}{3+\frac{1}{1+\frac{1}{\frac{2}{2}+\frac{1}{2}}}}$$
$$2+\frac{1}{3+\frac{1}{1+\frac{1}{\frac{2+1}{2}}}}$$
$$2+\frac{1}{3+\frac{1}{1+\frac{1}{\frac{3}{2}}}}$$
$$2+\frac{1}{3+\frac{1}{1+1\times \frac{2}{3}}}$$
$$2+\frac{1}{3+\frac{1}{1+\frac{2}{3}}}$$
$$2+\frac{1}{3+\frac{1}{\frac{3}{3}+\frac{2}{3}}}$$
$$2+\frac{1}{3+\frac{1}{\frac{3+2}{3}}}$$
$$2+\frac{1}{3+\frac{1}{\frac{5}{3}}}$$
$$2+\frac{1}{3+1\times \frac{3}{5}}$$
$$2+\frac{1}{3+\frac{3}{5}}$$
$$2+\frac{1}{\frac{15}{5}+\frac{3}{5}}$$
$$2+\frac{1}{\frac{15+3}{5}}$$
$$2+\frac{1}{\frac{18}{5}}$$
$$2+1\times \frac{5}{18}$$
$$2+\frac{5}{18}$$
$$\frac{36}{18}+\frac{5}{18}$$
$$\frac{36+5}{18}$$
$$\frac{41}{18}$$
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$\frac{41}{2 \cdot 3 ^ {2}} = 2\frac{5}{18} = 2.2777777777777777$
