Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 4 / 2E (Numbers and Sets)

Write the rational number as $r=p/q$ with coprime integers $p,q$ and $q>0$. Substituting into the monic polynomial and multiplying by $q^n$ givesAll terms except the first are divisible by $q$, so $q\mid p^n$. If $q>1$, it has a prime factor dividing $p^n$, and the Euclid lemma implies that this prime divides $p$, contradicting coprimality. Thus $q=1$, proving this special case of the rational root theorem:

The exponential series at $1$ givesFor every positive integer $q$, multiply by the factorial $q!$ and separate the integer part:Every term in $R_q$ is positive. Each denominator is at least $(q+1{)}^j$, and the inequality is strict when $j\ge 2$. Comparison with the geometric series therefore givesIf $e=p/q$ with integers $p$ and $q\ge 1$, then $q!e=p(q-1)!$ would be an integer. Since $I_q$ is also an integer, $R_q$ would be an integer strictly between zero and one, which is impossible. HenceThis proves the irrationality of e by using its rapidly decreasing series tail.