= Solution
For a <series> $\sum u_n$ with $u_n>0$, suppose $L=\lim_{n\to\infty}u_{n+1}/u_n$ exists, allowing $L=+\infty$. The <ratio test> says \b[the series converges if $L<1$ and diverges if $L>1$]. In the first case an eventual ratio bound by some $q<1$ compares the tail to a <geometric series>. In the second case an eventual ratio above some $q>1$ prevents the terms tending to zero.
\b[If $L=1$, the test is inconclusive]: both $\sum1/n$ and $\sum1/n^2$ have limiting ratio one, while the first diverges and the second converges. If the ratio has no limit, this particular limit formulation gives no conclusion.
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