Solve 1/1991+frac{1{1992}+1/1993+1/1994+1/1995+frac{1}{1996}+frac{1}{1997}+frac{1}{1998}+frac{1}{1999}+frac{1}{2000}}

$$\frac{ 1 }{ 1991+ \frac{ 1 }{ 1992 } + \frac{ 1 }{ 1993 } + \frac{ 1 }{ 1994 } + \frac{ 1 }{ 1995 } + \frac{ 1 }{ 1996 } + \frac{ 1 }{ 1997 } + \frac{ 1 }{ 1998 } + \frac{ 1 }{ 1999 } + \frac{ 1 }{ 2000 } }$$

Evaluate

$$\frac{87300971946752152873914000}{173816628788299314456742122907}\approx 0.000502259$$

Solution Steps

Convert 1991 to fraction \frac{3966072}{1992}.

$$\frac{1}{\frac{3966072}{1992}+\frac{1}{1992}+\frac{1}{1993}+\frac{1}{1994}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Since \frac{3966072}{1992} and \frac{1}{1992} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{3966072+1}{1992}+\frac{1}{1993}+\frac{1}{1994}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Add 3966072 and 1 to get 3966073.

$$\frac{1}{\frac{3966073}{1992}+\frac{1}{1993}+\frac{1}{1994}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Least common multiple of 1992 and 1993 is 3970056. Convert \frac{3966073}{1992} and \frac{1}{1993} to fractions with denominator 3970056.

$$\frac{1}{\frac{7904383489}{3970056}+\frac{1992}{3970056}+\frac{1}{1994}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Since \frac{7904383489}{3970056} and \frac{1992}{3970056} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{7904383489+1992}{3970056}+\frac{1}{1994}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Add 7904383489 and 1992 to get 7904385481.

$$\frac{1}{\frac{7904385481}{3970056}+\frac{1}{1994}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Least common multiple of 3970056 and 1994 is 3958145832. Convert \frac{7904385481}{3970056} and \frac{1}{1994} to fractions with denominator 3958145832.

$$\frac{1}{\frac{7880672324557}{3958145832}+\frac{1985028}{3958145832}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Since \frac{7880672324557}{3958145832} and \frac{1985028}{3958145832} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{7880672324557+1985028}{3958145832}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Add 7880672324557 and 1985028 to get 7880674309585.

$$\frac{1}{\frac{7880674309585}{3958145832}+\frac{1}{1995}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Least common multiple of 3958145832 and 1995 is 2632166978280. Convert \frac{7880674309585}{3958145832} and \frac{1}{1995} to fractions with denominator 2632166978280.

$$\frac{1}{\frac{5240648415874025}{2632166978280}+\frac{1319381944}{2632166978280}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Since \frac{5240648415874025}{2632166978280} and \frac{1319381944}{2632166978280} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{5240648415874025+1319381944}{2632166978280}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Add 5240648415874025 and 1319381944 to get 5240649735255969.

$$\frac{1}{\frac{5240649735255969}{2632166978280}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Reduce the fraction \frac{5240649735255969}{2632166978280} to lowest terms by extracting and canceling out 3.

$$\frac{1}{\frac{1746883245085323}{877388992760}+\frac{1}{1996}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Least common multiple of 877388992760 and 1996 is 437817107387240. Convert \frac{1746883245085323}{877388992760} and \frac{1}{1996} to fractions with denominator 437817107387240.

$$\frac{1}{\frac{871694739297576177}{437817107387240}+\frac{219347248190}{437817107387240}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Since \frac{871694739297576177}{437817107387240} and \frac{219347248190}{437817107387240} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{871694739297576177+219347248190}{437817107387240}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Add 871694739297576177 and 219347248190 to get 871694958644824367.

$$\frac{1}{\frac{871694958644824367}{437817107387240}+\frac{1}{1997}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Least common multiple of 437817107387240 and 1997 is 874320763452318280. Convert \frac{871694958644824367}{437817107387240} and \frac{1}{1997} to fractions with denominator 874320763452318280.

$$\frac{1}{\frac{1740774832413714260899}{874320763452318280}+\frac{437817107387240}{874320763452318280}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Since \frac{1740774832413714260899}{874320763452318280} and \frac{437817107387240}{874320763452318280} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{1740774832413714260899+437817107387240}{874320763452318280}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Add 1740774832413714260899 and 437817107387240 to get 1740775270230821648139.

$$\frac{1}{\frac{1740775270230821648139}{874320763452318280}+\frac{1}{1998}+\frac{1}{1999}+\frac{1}{2000}}$$

Least common multiple of 874320763452318280 and 1998 is 873446442688865961720. Convert \frac{1740775270230821648139}{874320763452318280} and \frac{1}{1998} to fractions with denominator 873446442688865961720.

$$\frac{1}{\frac{1739034494960590826490861}{873446442688865961720}+\frac{437160381726159140}{873446442688865961720}+\frac{1}{1999}+\frac{1}{2000}}$$

Since \frac{1739034494960590826490861}{873446442688865961720} and \frac{437160381726159140}{873446442688865961720} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{1739034494960590826490861+437160381726159140}{873446442688865961720}+\frac{1}{1999}+\frac{1}{2000}}$$

Add 1739034494960590826490861 and 437160381726159140 to get 1739034932120972552650001.

$$\frac{1}{\frac{1739034932120972552650001}{873446442688865961720}+\frac{1}{1999}+\frac{1}{2000}}$$

Least common multiple of 873446442688865961720 and 1999 is 1746019438935043057478280. Convert \frac{1739034932120972552650001}{873446442688865961720} and \frac{1}{1999} to fractions with denominator 1746019438935043057478280.

$$\frac{1}{\frac{3476330829309824132747351999}{1746019438935043057478280}+\frac{873446442688865961720}{1746019438935043057478280}+\frac{1}{2000}}$$

Since \frac{3476330829309824132747351999}{1746019438935043057478280} and \frac{873446442688865961720}{1746019438935043057478280} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{3476330829309824132747351999+873446442688865961720}{1746019438935043057478280}+\frac{1}{2000}}$$

Add 3476330829309824132747351999 and 873446442688865961720 to get 3476331702756266821613313719.

$$\frac{1}{\frac{3476331702756266821613313719}{1746019438935043057478280}+\frac{1}{2000}}$$

Least common multiple of 1746019438935043057478280 and 2000 is 87300971946752152873914000. Convert \frac{3476331702756266821613313719}{1746019438935043057478280} and \frac{1}{2000} to fractions with denominator 87300971946752152873914000.

$$\frac{1}{\frac{173816585137813341080665685950}{87300971946752152873914000}+\frac{43650485973376076436957}{87300971946752152873914000}}$$

Since \frac{173816585137813341080665685950}{87300971946752152873914000} and \frac{43650485973376076436957}{87300971946752152873914000} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{173816585137813341080665685950+43650485973376076436957}{87300971946752152873914000}}$$

Add 173816585137813341080665685950 and 43650485973376076436957 to get 173816628788299314456742122907.

$$\frac{1}{\frac{173816628788299314456742122907}{87300971946752152873914000}}$$

Divide 1 by \frac{173816628788299314456742122907}{87300971946752152873914000} by multiplying 1 by the reciprocal of \frac{173816628788299314456742122907}{87300971946752152873914000}.

$$1\times \frac{87300971946752152873914000}{173816628788299314456742122907}$$

Multiply 1 and \frac{87300971946752152873914000}{173816628788299314456742122907} to get \frac{87300971946752152873914000}{173816628788299314456742122907}.

$$\frac{87300971946752152873914000}{173816628788299314456742122907}$$

Factor

$$\frac{2 ^ {4} \cdot 3 ^ {3} \cdot 5 ^ {3} \cdot 7 \cdot 19 \cdot 37 \cdot 83 \cdot 499 \cdot 997 \cdot 1993 \cdot 1997 \cdot 1999}{173816628788299314456742122907} = 0.0005022590333004372$$