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Interesting trigonometric MRB constant approximations

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*)
POSTED BY: Marvin Ray Burns
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