Solve 1+1/1!+1/2!+1/3!+1/4!+frac{1}{5!}+frac{1}{6!}

$$1+ \frac{ 1 }{ 1 ! } + \frac{ 1 }{ 2 ! } + \frac{ 1 }{ 3 ! } + \frac{ 1 }{ 4 ! } + \frac{ 1 }{ 5 ! } + \frac{ 1 }{ 6 ! }$$

Evaluate

$$\frac{1957}{720}\approx 2.718055556$$

Solution Steps

The factorial of 1 is 1.

$$1+\frac{1}{1}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

Anything divided by one gives itself.

$$1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

Add 1 and 1 to get 2.

$$2+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

The factorial of 2 is 2.

$$2+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

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

$$\frac{4}{2}+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

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

$$\frac{4+1}{2}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

Add 4 and 1 to get 5.

$$\frac{5}{2}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

The factorial of 3 is 6.

$$\frac{5}{2}+\frac{1}{6}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

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

$$\frac{15}{6}+\frac{1}{6}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

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

$$\frac{15+1}{6}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

Add 15 and 1 to get 16.

$$\frac{16}{6}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

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

$$\frac{8}{3}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

The factorial of 4 is 24.

$$\frac{8}{3}+\frac{1}{24}+\frac{1}{5!}+\frac{1}{6!}$$

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

$$\frac{64}{24}+\frac{1}{24}+\frac{1}{5!}+\frac{1}{6!}$$

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

$$\frac{64+1}{24}+\frac{1}{5!}+\frac{1}{6!}$$

Add 64 and 1 to get 65.

$$\frac{65}{24}+\frac{1}{5!}+\frac{1}{6!}$$

The factorial of 5 is 120.

$$\frac{65}{24}+\frac{1}{120}+\frac{1}{6!}$$

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

$$\frac{325}{120}+\frac{1}{120}+\frac{1}{6!}$$

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

$$\frac{325+1}{120}+\frac{1}{6!}$$

Add 325 and 1 to get 326.

$$\frac{326}{120}+\frac{1}{6!}$$

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

$$\frac{163}{60}+\frac{1}{6!}$$

The factorial of 6 is 720.

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

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

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

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

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

Add 1956 and 1 to get 1957.

$$\frac{1957}{720}$$

Factor

$$\frac{19 \cdot 103}{2 ^ {4} \cdot 3 ^ {2} \cdot 5} = 2\frac{517}{720} = 2.7180555555555554$$