Why doesn't Mathematica provide an answer while Wolfram|Alpha does, concerning a series convergence?

Question

Among other series I've been working on, I was asked to find whether $$\sum_n 1-\cos(\frac{\pi}{n})$$ converged, and Mathematica's output to SumConvergence[1 - Cos[Pi/n], n] simply was repeating the input, without further information. Wolfram|Alpha, though, at least told me which test were or not conclusive.

I'm new to Mathematica, and even though I've looked both on Google and into Wolfram's documentation, I haven't found information that could help me figure out how to get, from Mathematica, the conditions for the convergence of a series involving something else than powers of a variable.

I would appreciate if you could give me some clues on the typical procedure to make Mathematica correctly evaluate the convergence of a series, or/and to return the conditions for convergence. Thank you in advance.

Answer 1

Here's a hack, which works in V13.0.1:

The Raabe test "fails" internally because if the summand contains a trigonometric function, which is explicitly tested for, the Raabe test is not tried. Also if a complex number is present in the integrand, then the Raabe test is not tried, so the following does not work:

SumConvergence[1 - Cos[Pi/n] // TrigToExp, n] (* FAILS *)

So we have to get even trickier:

a // ClearAll;
a[n_] /; MemberQ[Stack[], Limit] := 1 - Cos[Pi/n];
SumConvergence[a[n], n]
(*  True  *)

This hides the Cos until it is inside Limit.

I assume the trigonometric functions are excluded for the reason @Валерий Заподовников points out, that it is too difficult to determine the signs of the terms. For instance, this takes about 0.25, which is fairly slow for an easy problem:

ClearSystemCache[];
Simplify[
  Reduce[
   n > 10^5 \[Implies] 
     (1 - Cos[Pi/n]) Sign[1 - Cos[Pi/n] /. n -> 10^5] > 0,
   n],
  n ∈ Integers && n > 10^5] // AbsoluteTiming

(*  {0.244874, True}  *)

Answer 2

This is a bug in mathematica, still present in 13.0.1 not like in https://mathematica.stackexchange.com/a/259312/82985, there DivergenceTest fails (since older buggy version of Limit is used inside SumConvergence[]) and thus no other tests are applied. Here RaabeTest has a bug because (IMHO) it cannot check the sequence is nonnegative (indeed the cos term goes to 1 after x is 2, but changes sign there, may be not obvious that after doing "1-" it is all positive).

So doing Raabe test by hand allows to find (>1 is convergent):

b[n_] = 1 - Cos[Pi/n];

Limit[n (b[n]/b[n + 1] - 1), n -> Infinity] (* 2 *)

or even Bertrand test (it is used if previous test is 1 so useless, but can be used even if Raabe test is good, it will be just Infinity or -Infinity) (>1 is convergent).

Limit[Log[n] (n (b[n]/b[n + 1] - 1) - 1), n -> Infinity] (*Infinity*)

P.S.

SumConvergence[a[n], n, Method -> "RaabeTest"]

just returns unevaluated.

BTW, it is also quite bad that Mathematica does not have Bertrand test.