Solve 1+1+1/2+1/6+1/24+1/120+frac{1}{720}+frac{1}{5040}

$$1+1+ \frac{ 1 }{ 2 } + \frac{ 1 }{ 6 } + \frac{ 1 }{ 24 } + \frac{ 1 }{ 120 } + \frac{ 1 }{ 720 } + \frac{ 1 }{ 5040 }$$

Evaluate

$$\frac{685}{252}\approx 2.718253968$$

Solution Steps

Add 1 and 1 to get 2.

$$2+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Convert 2 to fraction \frac{4}{2}.

$$\frac{4}{2}+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Since \frac{4}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{4+1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Add 4 and 1 to get 5.

$$\frac{5}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Least common multiple of 2 and 6 is 6. Convert \frac{5}{2} and \frac{1}{6} to fractions with denominator 6.

$$\frac{15}{6}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Since \frac{15}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\frac{15+1}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Add 15 and 1 to get 16.

$$\frac{16}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Reduce the fraction \frac{16}{6} to lowest terms by extracting and canceling out 2.

$$\frac{8}{3}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Least common multiple of 3 and 24 is 24. Convert \frac{8}{3} and \frac{1}{24} to fractions with denominator 24.

$$\frac{64}{24}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Since \frac{64}{24} and \frac{1}{24} have the same denominator, add them by adding their numerators.

$$\frac{64+1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Add 64 and 1 to get 65.

$$\frac{65}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Least common multiple of 24 and 120 is 120. Convert \frac{65}{24} and \frac{1}{120} to fractions with denominator 120.

$$\frac{325}{120}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Since \frac{325}{120} and \frac{1}{120} have the same denominator, add them by adding their numerators.

$$\frac{325+1}{120}+\frac{1}{720}+\frac{1}{5040}$$

Add 325 and 1 to get 326.

$$\frac{326}{120}+\frac{1}{720}+\frac{1}{5040}$$

Reduce the fraction \frac{326}{120} to lowest terms by extracting and canceling out 2.

$$\frac{163}{60}+\frac{1}{720}+\frac{1}{5040}$$

Least common multiple of 60 and 720 is 720. Convert \frac{163}{60} and \frac{1}{720} to fractions with denominator 720.

$$\frac{1956}{720}+\frac{1}{720}+\frac{1}{5040}$$

Since \frac{1956}{720} and \frac{1}{720} have the same denominator, add them by adding their numerators.

$$\frac{1956+1}{720}+\frac{1}{5040}$$

Add 1956 and 1 to get 1957.

$$\frac{1957}{720}+\frac{1}{5040}$$

Least common multiple of 720 and 5040 is 5040. Convert \frac{1957}{720} and \frac{1}{5040} to fractions with denominator 5040.

$$\frac{13699}{5040}+\frac{1}{5040}$$

Since \frac{13699}{5040} and \frac{1}{5040} have the same denominator, add them by adding their numerators.

$$\frac{13699+1}{5040}$$

Add 13699 and 1 to get 13700.

$$\frac{13700}{5040}$$

Reduce the fraction \frac{13700}{5040} to lowest terms by extracting and canceling out 20.

$$\frac{685}{252}$$

Factor

$$\frac{5 \cdot 137}{2 ^ {2} \cdot 3 ^ {2} \cdot 7} = 2\frac{181}{252} = 2.7182539682539684$$