= Solution
For a positive-term <series>, the <ratio test> says that if $a_{n+1}/a_n\to L<1$, the series converges; if the ratio tends to $L>1$, including positive infinity, it diverges. More generally $\limsup a_{n+1}/a_n<1$ suffices for convergence and $\liminf a_{n+1}/a_n>1$ for divergence. \b[A limiting ratio of one gives no conclusion.]
The <P-series> $\sum n^{-2}$ converges and the <harmonic series> $\sum n^{-1}$ diverges. Both have strictly decreasing positive terms, and their successive ratios, $(n/(n+1))^2$ and $n/(n+1)$, tend to one. These supply the two requested possibilities.
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