= Solution
The <ratio test> uses $L=\lim_{n\to\infty}|a_{n+1}/a_n|$, when it exists for eventually nonzero terms: $L<1$ gives <absolute convergence>, $L>1$ gives divergence, and $L=1$ is inconclusive.
For $a_n=n/(2^n-1)$,
$$
\frac{a_{n+1}}{a_n}=\frac{n+1}{n}\frac{2^n-1}{2^{n+1}-1}\longrightarrow\frac12.
$$
Thus the first <series> \b[converges absolutely]. For the alternating second <series>,
$$
\left|\frac{a_{n+1}}{a_n}\right|=2\left(\frac n{n+1}\right)^5\longrightarrow2,
$$
so it \b[diverges]. Alternating signs cannot rescue terms whose <absolute values> fail to tend to zero.
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