Mister Exam
Least common denominator factorial(n)/factorial(n+1)-(n-1)/factorial(n)
The solution
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n! n - 1 -------- - ----- (n + 1)! n!
$$- \frac{n - 1}{n!} + \frac{n!}{\left(n + 1\right)!}$$
factorial(n)/factorial(n + 1) - (n - 1)/factorial(n)
Numerical answer
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factorial(n)/factorial(n + 1) - (-1.0 + n)/factorial(n)
factorial(n)/factorial(n + 1) - (-1.0 + n)/factorial(n)
Rational denominator
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2
n! - n*(1 + n)! + (1 + n)!
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n!*(1 + n)!
$$\frac{- n \left(n + 1\right)! + n!^{2} + \left(n + 1\right)!}{n! \left(n + 1\right)!}$$
(factorial(n)^2 - n*factorial(1 + n) + factorial(1 + n))/(factorial(n)*factorial(1 + n))
Common denominator
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/ 2 \
-\- n! - (1 + n)! + n*(1 + n)!/
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n!*(1 + n)!
$$- \frac{n \left(n + 1\right)! - n!^{2} - \left(n + 1\right)!}{n! \left(n + 1\right)!}$$
-(-factorial(n)^2 - factorial(1 + n) + n*factorial(1 + n))/(factorial(n)*factorial(1 + n))
Powers
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1 - n n! ----- + -------- n! (1 + n)!
$$\frac{1 - n}{n!} + \frac{n!}{\left(n + 1\right)!}$$
(1 - n)/factorial(n) + factorial(n)/factorial(1 + n)
Combinatorics
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-(-Gamma(n) + n*Gamma(1 + n) - Gamma(n)*Gamma(1 + n))
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Gamma(n)*Gamma(2 + n)
$$- \frac{n \Gamma\left(n + 1\right) - \Gamma\left(n\right) \Gamma\left(n + 1\right) - \Gamma\left(n\right)}{\Gamma\left(n\right) \Gamma\left(n + 2\right)}$$
-(-gamma(n) + n*gamma(1 + n) - gamma(n)*gamma(1 + n))/(gamma(n)*gamma(2 + n))
Combining rational expressions
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2
n! - (-1 + n)*(1 + n)!
-----------------------
n!*(1 + n)!
$$\frac{- \left(n - 1\right) \left(n + 1\right)! + n!^{2}}{n! \left(n + 1\right)!}$$
(factorial(n)^2 - (-1 + n)*factorial(1 + n))/(factorial(n)*factorial(1 + n))