Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 2F (Topics in Analysis)

The Liouville approximation theorem says that if $\alpha$ is an irrational algebraic number of degree $d$, there is $c_{\alpha }>0$ such that $|\alpha -p/q|\ge c_{\alpha }{q}^{-d}$ for all integers $p$ and $q\ge 1$. To see the source of the bound, a degree-$d$ integer minimal polynomial $P$ gives $q^{d}P(p/q)$ a nonzero integer. On a fixed neighbourhood of $\alpha$, the mean value theorem bounds $|P(p/q)|$ by a constant times $|p/q-\alpha |$; outside that neighbourhood a smaller uniform constant suffices.

Let $\alpha$ denote the factorial-denominator sum and let $p_N/q_N$ be its partial sum through $N$, with $q_N=10^{N!}$ for $N\ge 2$. This is an integer numerator even though $0!=1!=1$ makes the first two terms equal. Positivity and the rapid increase of factorial exponents implyFor example the first omitted term is $10^{-(N+1)!}$ and every succeeding factorial exponent increases by at least one, giving a geometric tail bound less than $10/9$ times that first term.

First $\alpha$ is irrational: if $\alpha =a/b$, the positive difference from a partial sum is at least $1/(b{q}_N)$, contradicting this bound for sufficiently large $N$. If it were algebraic of any fixed degree $d$, Liouville approximation theorem would require $c_{\alpha }{q}_N^{-d}<2q_N^{-(N+1)}$, again impossible for large $N$. Hence $\boxed{\alpha \text{ is transcendental}.}$