= Solution
The <ratio test> says that if
$$
\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=L<1,
$$
then $\sum a_n$ 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 $n$,
$$
|a_{n+1}|\leq q|a_n|.
$$
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 $|a_{n+1}|\geq q|a_n|$, so $a_n$ cannot tend to zero; the <series> therefore diverges.
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