In the discussion https://math.stackexchange.com/a/3419778/198592 I stumbled of the question how to calculate the sum
$$s= \sum _{n=3}^{\infty } \frac{n \cot \left(\frac{\pi }{n}\right)}{4^{n-2}}$$
to high precision.
I used NSum[]
and played around with WorkingPrecision
and Method
, and the best I could get was 29 digits:
NSum[(n Cot[\[Pi]/n])/4^(n - 2), {n, 3, \[Infinity]}, WorkingPrecision -> 29,
Method -> "WynnEpsilon"]
(* Out[123]= 0.85222988130298006255574122938 *)
Increasing WorkingPrecision
did not help as can be see from
sn[w_] := NSum[(n Cot[\[Pi]/n])/4^(n - 2), {n, 3, \[Infinity]},
WorkingPrecision -> w, Method -> "WynnEpsilon"]
Table[{w, Length[RealDigits[sn1[w]][[1]]]}, {w, 24, 32}]
(* Out[126]= {{24, 24}, {25, 24}, {26, 26}, {27, 26}, {28, 27}, {29, 29}, {30, 28}, {31,
28}, {32, 28}} *)
How can I get more exact digits, say 50 or 100?