The most fullproof test for convergence

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SUMMARY

The discussion centers on the search for the "most foolproof" test for convergence of series, specifically questioning the reliability of Kummer's test. Participants acknowledge that Kummer's test, which relies on comparison, cannot be deemed foolproof due to the existence of series that do not conform to a single standard of convergence or divergence. The conversation highlights the need for alternative tests, with a particular focus on Gauss's form of Kummer's test and the limitations of existing methods in addressing series that resist convergence tests.

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  • Understanding of Kummer's test for convergence
  • Familiarity with Gauss's form of Kummer's test
  • Knowledge of convergence and divergence in series
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Enjolras1789
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The "most fullproof" test for convergence

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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Enjolras1789 said:
I am not 100% sure of my meaning of "most fullproof."
Me neither. There is no one fits all solution. It has a reason why there are so many different tests.
 

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