4
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Looking into this question made me suspect that SumConvergence might have more Methods available than the four listed in its documentation. How do I find all Methods available for this function?

Edit: For completeness let me add that, besides Automatic, the documentation mentions (note the vague choice of words: "possible values ... include")

Possible values for Method include:
"IntegralTest" the integral test
"RaabeTest" Raabe's test
"RatioTest" D'Alembert ratio test
"RootTest" Cauchy root test

Note: The obvious adaption to the answer to this similar question does not seem to work; the same seems to be true for the answers to this question.

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    $\begingroup$ The code within Sum`SumConvergenceDump`SumConvergenceTestMethod[] seems to show the names of the supported methods. $\endgroup$ – J. M. will be back soon Dec 12 '16 at 13:17
  • $\begingroup$ @J.M: Probably I'm very naive, but if I evaluate that noting happens; I just get back the input. $\endgroup$ – Jules Lamers Dec 12 '16 at 13:22
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    $\begingroup$ You need to have run SumConvergence[] first before trying ??Sum`SumConvergenceDump`SumConvergenceTestMethod. $\endgroup$ – J. M. will be back soon Dec 12 '16 at 13:22
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    $\begingroup$ I also see "DivergenceTest" in the code. $\endgroup$ – J. M. will be back soon Dec 12 '16 at 13:31
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    $\begingroup$ Can't really say; that's what I got after cursory spelunking. Someone else who knows more than me might want to chime in instead. $\endgroup$ – J. M. will be back soon Dec 12 '16 at 13:39
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Like J.M., I see the "DivergenceTest" (Nth term test for divergence) as well as user-defined methods.

The "DivergenceTest" can only be used to show divergence (return value False). If it returns True, it is unreliable.

SumConvergence[1/n, n, Method -> "DivergenceTest"]
(*  True  *)

User-defined methods have the form Method -> myConvTest, where myConvTest has the form as SumConvergence (without options):

myConvTest[summand, variable]

Examples:

SumConvergence[1/n, n, Method -> (True &)] (* the optimistic test *)
(*  True  *)

myLCT[e_, k_] := SumConvergence[Normal[Series[e, {k, ∞, 2}]], k]
SumConvergence[Sin[50/n^2], n, Method -> myLCT]
(*  True  *)

[Discovered exploring this question.]

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  • $\begingroup$ Thanks, that is a useful addition to J.M.'s comments! $\endgroup$ – Jules Lamers Jan 13 '18 at 15:21

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