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What is the "most foolproof" test for convergence of a series? And in my question, I am not 100% sure of my meaning of "most fullproof."

I understand that any test based on Kummer's test can never be foolproof...Kummer's test is based on a comparison test, and a proof exists that there is no "most slowly" converging/diverging series against which to compare. What is the "best" standard test you would use, off the top of your head, if you had to do one? Would you use Gauss's form of Kummer's test by default, or something else?

Are any theorists (I am thinking string theorists of various forms) familiar with series which resist convergence tests do-able by Gauss's test? I am not asking if one is familiar with divergent series here, simply series which are not clearly establishable as convergent or divergent by a Kummer test?

Is there a more sensitive test to do than a Kummer test?

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# The most fullproof test for convergence

Can you offer guidance or do you also need help?

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