A rational number that is a infinite product of distinct irrational numbers?

perlejatyh8

perlejatyh8

Answered question

2022-11-19

A rational number that is a infinite product of distinct irrational numbers?

Answer & Explanation

trivialaxxf

trivialaxxf

Beginner2022-11-20Added 21 answers

Let c be any nonzero rational number.
Let P n be the n th th prime number.
Define a sequence of nonzero rational numbers r n recursively so that
r 1 r 2 r n = c P 1 P 2 P n .
Let a n = r n P n . Then
lim n a 1 a 2 a n = lim n c P 1 P 2 P n P 1 P 2 P n = c .
Thus the infinite product n = 1 a n converges to c. The product of any nonempty finite set of distinct terms is irrational, being a nonzero rational multiple of the square root of a product of distinct primes.

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