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

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

Evaluate

$$\frac{1237}{720}\approx 1.718055556$$

Solution Steps

The factorial of 2 is 2.

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

Convert 1 to fraction \frac{2}{2}.

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

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

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

Add 2 and 1 to get 3.

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

The factorial of 3 is 6.

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

Least common multiple of 2 and 6 is 6. Convert \frac{3}{2} and \frac{1}{6} to fractions with denominator 6.

$$\frac{9}{6}+\frac{1}{6}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

Since \frac{9}{6} and \frac{1}{6} have the same denominator, add them by adding their numerators.

$$\frac{9+1}{6}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

Add 9 and 1 to get 10.

$$\frac{10}{6}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!}$$

Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.

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

The factorial of 4 is 24.

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

Least common multiple of 3 and 24 is 24. Convert \frac{5}{3} and \frac{1}{24} to fractions with denominator 24.

$$\frac{40}{24}+\frac{1}{24}+\frac{1}{5!}+\frac{1}{6!}$$

Since \frac{40}{24} and \frac{1}{24} have the same denominator, add them by adding their numerators.

$$\frac{40+1}{24}+\frac{1}{5!}+\frac{1}{6!}$$

Add 40 and 1 to get 41.

$$\frac{41}{24}+\frac{1}{5!}+\frac{1}{6!}$$

The factorial of 5 is 120.

$$\frac{41}{24}+\frac{1}{120}+\frac{1}{6!}$$

Least common multiple of 24 and 120 is 120. Convert \frac{41}{24} and \frac{1}{120} to fractions with denominator 120.

$$\frac{205}{120}+\frac{1}{120}+\frac{1}{6!}$$

Since \frac{205}{120} and \frac{1}{120} have the same denominator, add them by adding their numerators.

$$\frac{205+1}{120}+\frac{1}{6!}$$

Add 205 and 1 to get 206.

$$\frac{206}{120}+\frac{1}{6!}$$

Reduce the fraction \frac{206}{120} to lowest terms by extracting and canceling out 2.

$$\frac{103}{60}+\frac{1}{6!}$$

The factorial of 6 is 720.

$$\frac{103}{60}+\frac{1}{720}$$

Least common multiple of 60 and 720 is 720. Convert \frac{103}{60} and \frac{1}{720} to fractions with denominator 720.

$$\frac{1236}{720}+\frac{1}{720}$$

Since \frac{1236}{720} and \frac{1}{720} have the same denominator, add them by adding their numerators.

$$\frac{1236+1}{720}$$

Add 1236 and 1 to get 1237.

$$\frac{1237}{720}$$

Factor

$$\frac{1237}{2 ^ {4} \cdot 3 ^ {2} \cdot 5} = 1\frac{517}{720} = 1.7180555555555554$$