Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 9F / i / Solution

The ratio test says that ifthen $\sum a_n$ converges absolutely. If the ratio has a limit $L>1$, the series diverges.

For the first assertion, choose $q$ with $L<q<1$. For all sufficiently large $n$,Iteration bounds the tail by a constant multiple of the convergent geometric series $\sum q^n$. If $L>1$, choose $q$ with $1<q<L$. Eventually $|a_{n+1}|\ge q|a_n|$, so $a_n$ cannot tend to zero; the series therefore diverges.