Solve 2+1/1+1{2+1/1+1{1+frac{1/4+frac{1{1+frac{1/1+frac{1{6+frac{1}{1+frac{1}{1+frac{1}{8+frac{1}{1+frac{1}{1+frac{1}{10}}}}}}}}}}}}}}

$$2+ \frac{ 1 }{ 1+ \frac{ 1 }{ 2+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 4+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 6+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 8+ \frac{ 1 }{ 1+ \frac{ 1 }{ 1+ \frac{ 1 }{ 10 } } } } } } } } } } } } } }$$

Evaluate

$$\frac{517656}{190435}\approx 2.718281828$$

Solution Steps

Convert 1 to fraction \frac{10}{10}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{1}{1+\frac{1}{\frac{10}{10}+\frac{1}{10}}}}}}}}}}}}}}$$

Since \frac{10}{10} and \frac{1}{10} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{1}{1+\frac{1}{\frac{10+1}{10}}}}}}}}}}}}}}$$

Add 10 and 1 to get 11.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{1}{1+\frac{1}{\frac{11}{10}}}}}}}}}}}}}}$$

Divide 1 by \frac{11}{10} by multiplying 1 by the reciprocal of \frac{11}{10}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{1}{1+1\times \frac{10}{11}}}}}}}}}}}}}$$

Multiply 1 and \frac{10}{11} to get \frac{10}{11}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{1}{1+\frac{10}{11}}}}}}}}}}}}}$$

Convert 1 to fraction \frac{11}{11}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{1}{\frac{11}{11}+\frac{10}{11}}}}}}}}}}}}}$$

Since \frac{11}{11} and \frac{10}{11} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{1}{\frac{11+10}{11}}}}}}}}}}}}}$$

Add 11 and 10 to get 21.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{1}{\frac{21}{11}}}}}}}}}}}}}$$

Divide 1 by \frac{21}{11} by multiplying 1 by the reciprocal of \frac{21}{11}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+1\times \frac{11}{21}}}}}}}}}}}}$$

Multiply 1 and \frac{11}{21} to get \frac{11}{21}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{8+\frac{11}{21}}}}}}}}}}}}$$

Convert 8 to fraction \frac{168}{21}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{168}{21}+\frac{11}{21}}}}}}}}}}}}$$

Since \frac{168}{21} and \frac{11}{21} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{168+11}{21}}}}}}}}}}}}$$

Add 168 and 11 to get 179.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{179}{21}}}}}}}}}}}}$$

Divide 1 by \frac{179}{21} by multiplying 1 by the reciprocal of \frac{179}{21}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+1\times \frac{21}{179}}}}}}}}}}}$$

Multiply 1 and \frac{21}{179} to get \frac{21}{179}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{1+\frac{21}{179}}}}}}}}}}}$$

Convert 1 to fraction \frac{179}{179}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{\frac{179}{179}+\frac{21}{179}}}}}}}}}}}$$

Since \frac{179}{179} and \frac{21}{179} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{\frac{179+21}{179}}}}}}}}}}}$$

Add 179 and 21 to get 200.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{1}{\frac{200}{179}}}}}}}}}}}$$

Divide 1 by \frac{200}{179} by multiplying 1 by the reciprocal of \frac{200}{179}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+1\times \frac{179}{200}}}}}}}}}}$$

Multiply 1 and \frac{179}{200} to get \frac{179}{200}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{1+\frac{179}{200}}}}}}}}}}$$

Convert 1 to fraction \frac{200}{200}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{\frac{200}{200}+\frac{179}{200}}}}}}}}}}$$

Since \frac{200}{200} and \frac{179}{200} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{\frac{200+179}{200}}}}}}}}}}$$

Add 200 and 179 to get 379.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{1}{\frac{379}{200}}}}}}}}}}$$

Divide 1 by \frac{379}{200} by multiplying 1 by the reciprocal of \frac{379}{200}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+1\times \frac{200}{379}}}}}}}}}$$

Multiply 1 and \frac{200}{379} to get \frac{200}{379}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{6+\frac{200}{379}}}}}}}}}$$

Convert 6 to fraction \frac{2274}{379}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{2274}{379}+\frac{200}{379}}}}}}}}}$$

Since \frac{2274}{379} and \frac{200}{379} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{2274+200}{379}}}}}}}}}$$

Add 2274 and 200 to get 2474.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{2474}{379}}}}}}}}}$$

Divide 1 by \frac{2474}{379} by multiplying 1 by the reciprocal of \frac{2474}{379}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+1\times \frac{379}{2474}}}}}}}}$$

Multiply 1 and \frac{379}{2474} to get \frac{379}{2474}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{1+\frac{379}{2474}}}}}}}}$$

Convert 1 to fraction \frac{2474}{2474}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{\frac{2474}{2474}+\frac{379}{2474}}}}}}}}$$

Since \frac{2474}{2474} and \frac{379}{2474} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{\frac{2474+379}{2474}}}}}}}}$$

Add 2474 and 379 to get 2853.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{1}{\frac{2853}{2474}}}}}}}}$$

Divide 1 by \frac{2853}{2474} by multiplying 1 by the reciprocal of \frac{2853}{2474}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+1\times \frac{2474}{2853}}}}}}}$$

Multiply 1 and \frac{2474}{2853} to get \frac{2474}{2853}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{1+\frac{2474}{2853}}}}}}}$$

Convert 1 to fraction \frac{2853}{2853}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{\frac{2853}{2853}+\frac{2474}{2853}}}}}}}$$

Since \frac{2853}{2853} and \frac{2474}{2853} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{\frac{2853+2474}{2853}}}}}}}$$

Add 2853 and 2474 to get 5327.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{1}{\frac{5327}{2853}}}}}}}$$

Divide 1 by \frac{5327}{2853} by multiplying 1 by the reciprocal of \frac{5327}{2853}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+1\times \frac{2853}{5327}}}}}}$$

Multiply 1 and \frac{2853}{5327} to get \frac{2853}{5327}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{4+\frac{2853}{5327}}}}}}$$

Convert 4 to fraction \frac{21308}{5327}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{21308}{5327}+\frac{2853}{5327}}}}}}$$

Since \frac{21308}{5327} and \frac{2853}{5327} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{21308+2853}{5327}}}}}}$$

Add 21308 and 2853 to get 24161.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{24161}{5327}}}}}}$$

Divide 1 by \frac{24161}{5327} by multiplying 1 by the reciprocal of \frac{24161}{5327}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+1\times \frac{5327}{24161}}}}}$$

Multiply 1 and \frac{5327}{24161} to get \frac{5327}{24161}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{1+\frac{5327}{24161}}}}}$$

Convert 1 to fraction \frac{24161}{24161}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{\frac{24161}{24161}+\frac{5327}{24161}}}}}$$

Since \frac{24161}{24161} and \frac{5327}{24161} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{\frac{24161+5327}{24161}}}}}$$

Add 24161 and 5327 to get 29488.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{\frac{29488}{24161}}}}}$$

Divide 1 by \frac{29488}{24161} by multiplying 1 by the reciprocal of \frac{29488}{24161}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+1\times \frac{24161}{29488}}}}$$

Multiply 1 and \frac{24161}{29488} to get \frac{24161}{29488}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{24161}{29488}}}}$$

Convert 1 to fraction \frac{29488}{29488}.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{\frac{29488}{29488}+\frac{24161}{29488}}}}$$

Since \frac{29488}{29488} and \frac{24161}{29488} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{\frac{29488+24161}{29488}}}}$$

Add 29488 and 24161 to get 53649.

$$2+\frac{1}{1+\frac{1}{2+\frac{1}{\frac{53649}{29488}}}}$$

Divide 1 by \frac{53649}{29488} by multiplying 1 by the reciprocal of \frac{53649}{29488}.

$$2+\frac{1}{1+\frac{1}{2+1\times \frac{29488}{53649}}}$$

Multiply 1 and \frac{29488}{53649} to get \frac{29488}{53649}.

$$2+\frac{1}{1+\frac{1}{2+\frac{29488}{53649}}}$$

Convert 2 to fraction \frac{107298}{53649}.

$$2+\frac{1}{1+\frac{1}{\frac{107298}{53649}+\frac{29488}{53649}}}$$

Since \frac{107298}{53649} and \frac{29488}{53649} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{1+\frac{1}{\frac{107298+29488}{53649}}}$$

Add 107298 and 29488 to get 136786.

$$2+\frac{1}{1+\frac{1}{\frac{136786}{53649}}}$$

Divide 1 by \frac{136786}{53649} by multiplying 1 by the reciprocal of \frac{136786}{53649}.

$$2+\frac{1}{1+1\times \frac{53649}{136786}}$$

Multiply 1 and \frac{53649}{136786} to get \frac{53649}{136786}.

$$2+\frac{1}{1+\frac{53649}{136786}}$$

Convert 1 to fraction \frac{136786}{136786}.

$$2+\frac{1}{\frac{136786}{136786}+\frac{53649}{136786}}$$

Since \frac{136786}{136786} and \frac{53649}{136786} have the same denominator, add them by adding their numerators.

$$2+\frac{1}{\frac{136786+53649}{136786}}$$

Add 136786 and 53649 to get 190435.

$$2+\frac{1}{\frac{190435}{136786}}$$

Divide 1 by \frac{190435}{136786} by multiplying 1 by the reciprocal of \frac{190435}{136786}.

$$2+1\times \frac{136786}{190435}$$

Multiply 1 and \frac{136786}{190435} to get \frac{136786}{190435}.

$$2+\frac{136786}{190435}$$

Convert 2 to fraction \frac{380870}{190435}.

$$\frac{380870}{190435}+\frac{136786}{190435}$$

Since \frac{380870}{190435} and \frac{136786}{190435} have the same denominator, add them by adding their numerators.

$$\frac{380870+136786}{190435}$$

Add 380870 and 136786 to get 517656.

$$\frac{517656}{190435}$$

Factor

$$\frac{2 ^ {3} \cdot 3 \cdot 21569}{5 \cdot 7 \cdot 5441} = 2\frac{136786}{190435} = 2.7182818284454013$$