Apologies if the question is too broad or if it has been answered before, but I understand what makes a comparison test convergent and divergent, but what makes a comparison test inconclusive? I know that if the smaller series is divergent, then the larger one is also divergent. And if the larger series is convergent, then the smaller one is too. But what makes a comparison test inconclusive?
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$\begingroup$Any test is a comparison with some known series (or class of series). So for example the ratio test is about checking if your series is below a convergent geometric series or above a divergent geometric series. When the limit in that test is $1$, it means that your series is, in some sense, in the gap between all convergent geometric series and all divergent geometric series, and you need a more sensitive test to draw a conclusion.
So for example Raabe's test can be conclusive when $\lim a_{n+1}/a_n=1$ if the difference $\lim a_{n}/a_{n+1}-1$ is of the order of $1/n$, and identifies a generalized harmonic series $\sum 1/n^p$ that behaves asymptotically as the given series. There are many other such tests.
The key idea here is that given a convergent series, there is always another convergent series that converges more slowly, and given a divergent series, there is another series that diverges, its terms decaying faster.
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