Now I would not say this Raabe approach is always perfect. Limit (n (b/b - 1) - 1), n -> Infinity] (* Infinity *) It should be noted, that if the calculator finds sum. EDIT: interesting phenomenon: (n3D1)5E7000+tan (pi2F25En). Our online calculator, build on Wolfram Alpha system is able to test convergence of different series. But as does not know any closed form, does a numerical approximation. Raabe by hand again prints 2 while Betrand's prints Infinity. Possible explication: Wolfram Alpha applies some convergence test and says 'is convergent'. SumConvergence, n, Method -> "IntegralTest"] (* error *) SumConvergence, n, Method -> "DivergenceTest"] (* true, so useless *) SumConvergence, n, Method -> "RootTest"] (* error *) SumConvergence, n, Method -> "RatioTest"] (* error *) A test to determine if a given series converges or diverges. SumConvergence, n, Method -> "RaabeTest"] (* error *) SumConvergence, n, Method -> Automatic] (* error *) It is actually quite interesting, see: and related questions. Limit (n (b/b - 1) - 1), n -> Infinity] (* prints Infinity, so convergent *) Now, indeed both Raabe test by hand and next level of it (in the series of Kummer's tests), Bertrand test (strange it is not one of Methods, WTF, one can even use Extended Betrand that is continuation further of that): b = (1 - Log/n)^(2 n) FreeQ does not work on (1 - Log/n)^(2 n) though by myLCT does. Now, they broke IntegralTest long time ago, see SumConvergence difficulty first workaround on 1- Cos there does not help this case though, it errors out after some time but FreeQ idea and myLCT do work, WOW. SumConvergence, n, Method -> "IntegralTest"] (* Fails, infinite loop bug!! *) SumConvergence, n, Method -> "DivergenceTest"] (* False, a bug! *) SumConvergence, n, Method -> "RootTest"] (* Error *) SumConvergence, n, Method -> "RatioTest"] (* Error *) SumConvergence, n, Method -> "RaabeTest"] (* True, NICE. SumConvergence, n, Method -> Automatic] (* False, bug: does not TRY Raabe *) infinite sum calculator with steps Infinite Series - Math is Fun. Or the problem there is that Raabe is done on nonnegative series and it somehow fails to check it. Radius of convergence xnn, n wolframalpha radius of convergence xnn. First of all even if on this limit Raabe test Method does work there is worse example here: Why doesn't Mathematica provide an answer while Wolfram|Alpha does, concerning a series convergence? I suppose DivergenceTest has higher priority (remember it is only meaningful for False, for True it means nothing). This looks like a horrible bug present in 12.3.1 (still happens on 13.0.1) (indeed due to "SumConvergence uses pre V11.2 Limit" and it is used for DivergenceTest).
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