Proof by Contradiction:

Assumption: Let's assume that there exists a positive integer n such that Г(n-1) + Г(n+1) is rational.

Properties of Gamma Function:
- The gamma function Г(n) is defined as Г(n) = (n-1)! for all positive integers n.
- Since n is a positive integer, n-1 is also a positive integer.

Expanding the Expression:
- Г(n-1) + Г(n+1) = (n-2)! + n!

Writing n! in Terms of (n-2)!:
- n! = n * (n-1)!
- Substituting this in the expression, we get:
(n-2)! + n * (n-1)! = (n-2)! * (1 + n)

Reasoning:
- Since n is a positive integer, n+1 is also a positive integer.
- Therefore, (n-2)! and (1+n) are both positive integers.

Conclusion:
- The product of two positive integers is always a positive integer.
- Hence, (n-2)! * (1+n) is a positive integer.

Contradiction:
- We assumed that Г(n-1) + Г(n+1) is rational.
- However, we have shown that (n-2)! * (1+n) is a positive integer.
- Since a positive integer is not a rational number, we have reached a contradiction.

Final Verdict:
- Therefore, our initial assumption that there exists a positive integer n for which Г(n-1) + Г(n+1) is rational is false.
- Hence, there is no positive integer n that satisfies this condition.