If I know that a series
$\displaystyle
\sum a_n
$
is convergent. Is it not possible to further deduce that
$\displaystyle
\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n} \leq 1
$
Hjörtur
The implication in the other direction is straightforward: it's the Ratio Test:
Ratio Test - ProofWiki
Does that help at all?