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Find if A/B is irrational

Source: 2019 Jozsef Wildt International Math Competition-W. 22

May 18, 2020

Gamma functionnumber theory

Problem Statement

Let

A

and

B

the series:

A=\sum \limits_{n=1}^{\infty}\frac{C_{2n}^1}{C_{2n}^0+C_{2n}^1+\cdots +C_{2n}^{2n}},\ B=\sum \limits_{n=1}^{\infty}\frac{\Gamma \left(n+\frac{1}{2}\right) }{\Gamma \left(n+\frac{5}{2}\right)}

Study if

\frac{A}{B}

is irrational number.