Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 1 / 2F (Topics in Analysis)

The hyperbolic cosine has the absolutely convergent expansion $\cosh (1/2)={\sum }_{k\ge 0}1/[4^k(2k)!]$. Suppose its value were a rational $a/b$, with $b>0$. For sufficiently large $N$, $b\mid (2N)!$. Multiplying by $4^N(2N)!$ makes the assumed value an integer, and every term of the partial sum through $k=N$ is also an integer. Their difference is the positive remainderLet $q=[4(2N+1)(2N+2){]}^{-1}$. Every new pair of factorial factors is at least $(2N+1)(2N+2)$, so $0<R_N\le {\sum }_{j\ge 1}{q}^j=q/(1-q)<1$ for $N\ge 1$. No integer lies strictly between zero and one. This contradiction proves $\cosh (1/2)$ is irrational. The argument uses a finite integer partial sum and a controlled tail; it does not infer irrationality merely from an infinite expansion.