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Is there any simple series where all the usual convergence tests are inconclusive? And how is convergence/divergence determined for these series?

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    Sure, but this is a bit like asking "what's the largest number anyone has ever used?" If you use that number, you can always increment it. Create a series all the known tests fail for and someone will create a new test to deal with that series.2012-06-24
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    @RyanBudney Unless of course, creating a test to deal with that series is extremely difficult. I think this is a good question.2012-06-24
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    By usual, I mean http://en.wikipedia.org/wiki/Convergence_tests2012-06-24
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    It would be more interesting to know if there is some positive monotonically decreasing series. And where its not an open problem.2012-06-24

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