"game" - to calculate 1.00...001 ^^ 0.5 = ?

[andydude]

Oh Mathematica I can do

Code:

CarlemanMatrix[series:SeriesData[x_, x0_, _, _, n_, 1]] :=
  Table[D[Normal[series]^j, {x, k}]/k! /. {x -> x0}, {j, 0, n}, {k, 0, n}];
RegularIterate[series:SeriesData[x_, x0_, _, _, n_, 1], t_] :=
  SeriesData[x, x0, MatrixPower[CarlemanMatrix[series], t][[2]], 0, n, 1];

Then you can do regular tetration with this code:

Code:

tet = c + Normal[RegularIterate[Series[c^((x + c)/c) - c, {x, 0, 3}], y]]/.x->(1-c);
tet /. {c -> 2, y -> 0.5}

which would calculate \( {}^{0.5}(\sqrt{2}) \).