= Factorial-tail irrationality proof for a hyperbolic cosine
For a fixed positive integer $q$, $\cosh(1/q)=\sum_{k\ge0}q^{-2k}/(2k)!$ is irrational. If it were rational with denominator $b$, choose $N$ large enough that $b\mid(2N)!$. After multiplication by $q^{2N}(2N)!$, the finite partial sum is an integer and the remaining positive tail is bounded by a geometric series whose ratio is at most $1/[q^2(2N+1)(2N+2)]$. For sufficiently large $N$, this tail is strictly between zero and one, contradicting its integer value. Only a finite integer partial sum is used; convergence of an infinite series of rationals does not itself prove irrationality.
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