= Solution
Rearranging the assumed positive-term ratio inequality gives
$$
\frac{a_{n+1}}{b_{n+1}}\leq\frac{a_n}{b_n}.
$$
Induction therefore yields $a_n/b_n\leq a_1/b_1$ for every $n\geq1$. In particular
$$
0\leq\sum_{n=1}^N a_n
\leq\frac{a_1}{b_1}\sum_{n=1}^N b_n
\leq\frac{a_1}{b_1}\sum_{n=1}^{\infty}b_n.
$$
The positive <partial sums> of the first <series> are nondecreasing and bounded above, so the <bounded monotone sequence theorem> proves that they converge. This is the <ratio comparison test>, and establishes \b[convergence of $\sum a_n$].
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