How to find all methods available to SumConvergence?

Question

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.

The code within Sum`SumConvergenceDump`SumConvergenceTestMethod[] seems to show the names of the supported methods. — J. M.'s missing motivation, Dec 12 '16 at 13:17
@J.M: Probably I'm very naive, but if I evaluate that noting happens; I just get back the input. — Jules Lamers, Dec 12 '16 at 13:22
You need to have run SumConvergence[] first before trying ??Sum`SumConvergenceDump`SumConvergenceTestMethod. — J. M.'s missing motivation, Dec 12 '16 at 13:22
Thanks. Can we then be sure there are no further methods available? If so, please turn your comments into an answer that I will gladly accept. — Jules Lamers, Dec 12 '16 at 13:37
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. — J. M.'s missing motivation, Dec 12 '16 at 13:39

score 3 · Answer 1 · answered Jan 09 '18 at 22:42

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.]

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

How to find all methods available to SumConvergence?

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