= Solution
Apply the <ratio test> to the infinite series whose <partial sums> are $S_N$. Assuming the terms are eventually nonzero and the limit exists, let
$$
\ell=\lim_{k\to\infty}\left|\frac{u_{k+1}}{u_k}\right|.
$$
If \b[$\ell<1$], the series has <absolute convergence>, by eventual comparison with a decaying <geometric series>. If \b[$\ell>1$], including an infinite limit, the terms fail to tend to zero and the series diverges. If \b[$\ell=1$], the test is inconclusive: both $\sum1/k$ and $\sum1/k^2$ have that ratio limit, but only the second converges. A finite <partial sum> itself requires no convergence test; the issue is the limit as $N\to\infty$.
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