How to judge whether the series is absolutely convergent?

Question

I need to judge whether the series $\sum_{n=1}^{\infty}(-1)^{n}\left(1-\cos \frac{\alpha}{n}\right)$ (α>0) is absolutely convergent.

SumConvergence[(-1)^n (1 - Cos[α/n]), n]
SumConvergence[Abs[(1 - Cos[α/n])], n, Method -> Automatic]
SumConvergence[Abs[(1 - Cos[1/n])], n, Method -> Automatic]

But the above code can not determine whether the series is absolutely convergent. How can I solve this problem?

Answer 1

We had to calculate by hand.

1-Cos[1/n]==2 Sin[1/(2 n)]^2//Simplify
(* True *)
 Limit[2 Sin[1/(2 n)]^2/(1/n^2), n -> Infinity]
 (* 1/2 *)
 SumConvergence[1/n^2, n]
  (* True *) 

Answer 2

Making use of the limit comparison test, we have

SumConvergence[Abs[Normal[Series[(-1)^n*(1 - Cos[a/n]), {n, Infinity, 2}]]], n]
(*True*)