# How to judge whether the series is absolutely convergent?

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?

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 *)


Making use of the limit comparison test, we have

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

• It's a great way, and it's universal. Sep 1, 2020 at 4:44