= Solution
The <comparison test for series> states that if $0\leq a_n\leq b_n$ for all sufficiently large $n$, then convergence of $\sum b_n$ implies convergence of $\sum a_n$; conversely, divergence of $\sum a_n$ implies divergence of $\sum b_n$. A summable bound on $|a_n|$ proves <absolute convergence> of $\sum a_n$.
For the <ratio test>, suppose the terms are eventually nonzero and $L=\lim_{n\to\infty}|a_{n+1}/a_n|$ exists. If $L<1$, the <series> converges absolutely. If $L>1$, including $L=\infty$, the terms fail to tend to zero and the <series> diverges. If $L=1$, the test is inconclusive. An eventually zero <series> converges separately. Convergence means that the sequence of partial sums has a finite limit; the terms tending to zero is only a necessary condition for a <convergent series>.
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