Let m be the MRB constant to 40 digits of precision:
m = NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 40];
. After that
1 - Cos[m/119595162821256388427204517274628339609] - Pi^2/8*10^-(2*40 - 2)
gives 0.*10^-116.
In general,
magic = 2 m/Pi
(* 0.119595162821256388427204517274628339609*)
. After that
Table[
1 - Cos[m/Floor[magic*10^n]] - Pi^2/8*10^-(2 n), {n, 10, 40}]
(* {4.385464676791947466761321772*10^-30,
2.59208994291255051050593254*10^-33,
5.2896210219556330067818305*10^-36,
1.1633653405789904134448983*10^-38,
1.318014202380040427023339*10^-41, 8.0137497971723466567979*10^-45,
1.8243662751312499138693*10^-47, 1.738640025869652510552*10^-50,
8.8137753325369667176*10^-54, 5.612639698155533691*10^-57,
1.486384016794827039*10^-59, 4.2194528318579711*10^-63,
9.31971504972644*10^-67, 9.31971504972644*10^-69,
1.06720368700503*10^-71, 3.563976666485*10^-75,
1.500848825805*10^-77, 5.6659337329*10^-81, 1.5396780515*10^-83,
9.54885630*10^-87, 1.29634494*10^-89, 5.846824*10^-93,
1.720568*10^-95, 7.0066*10^-99, 8.172*10^-102, 1.982*10^-104,
1.26*10^-107, 2.*10^-111, 2.*10^-113, 0.*10^-116, 0.*10^-118}*)
Having a term that adds accuracy is not unique to magic = 2 m/Pi
and isn't even unique to the MRB constant :
m = NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 80];
1 - Cos[m/10^30] - 5*m^2*10^-61
(* -5.18946663509688*10^-125*)
e = N[E, 80];
1 - Cos[e/10^30] - 5*e^2*10^-61
(*-2.274922918047676628*10^-120*)