The most fullproof test for convergence

In summary, the conversation discusses the concept of a "most foolproof" test for convergence of a series, with a focus on Kummer's test and its limitations. The question is posed if there is a better alternative or a more sensitive test available, and if any theorists have encountered series that cannot be conclusively determined as convergent or divergent by Kummer's test. Ultimately, it is acknowledged that there is no single perfect solution and different tests may be more suitable depending on the series in question.
  • #1
Enjolras1789
52
2
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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  • #2
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.
 

FAQ: The most fullproof test for convergence

What is the most fullproof test for convergence?

The most fullproof test for convergence is the ratio test, also known as the D'Alembert's ratio test. It is a powerful tool used to determine if a series converges or diverges.

How does the ratio test work?

The ratio test works by comparing the absolute value of the ratio of consecutive terms in a series to a limiting value, known as the critical ratio. If the critical ratio is less than 1, the series converges. If it is greater than 1, the series diverges. If it is equal to 1, the test is inconclusive and another test must be used.

What types of series can the ratio test be applied to?

The ratio test can be applied to infinite series with positive terms. It cannot be used on series with negative terms or alternating series.

Are there any limitations to the ratio test?

Yes, there are some series for which the ratio test cannot determine convergence or divergence. These include series with alternating terms, series with zero terms, and series with terms that do not approach zero as n approaches infinity.

How does the ratio test compare to other convergence tests?

The ratio test is one of the most fullproof tests for convergence, as it can be used to determine the convergence or divergence of a wide range of series. However, it may not always be the most efficient test and other tests may be more suitable for specific types of series.

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