The terms of the series
(* first 100 terms of the series:*)
m100 = Table[N[(-1)^k (k^(1/k) - 1), 200], {k, 1, 100}];
whose partial sums give raise the the MRB constant (see for example in this community Record breaking direct summation of MRB constant ) contain the k-th root of k.
As always the question about all the other roots in the complex plane comes up:
Remove[burnsStrip]
burnsStrip[k_?NumericQ, w_?NumericQ] :=
Block[{m = RotationTransform[k, {-1, 0}]},
Polygon[{m[{-5/2, -w}], m[{1/2, -w}], m[{1/2, w}], m[{-5/2, w}]}]
]
Graphics[{{Green, burnsStrip[0, 1/40]}, {Yellow,
Sequence @@ (burnsStrip[#,
1/40] & /@ (Degree {143.8, 120, 90, 60, 36.2}))},
Table[Point /@
Evaluate[
ReIm[(-1)^k ((Flatten[Evaluate[Block[{x}, Solve[x^k == k, x]]]])[[All, 2]] - 1)]], {k, 1, 217}]},
Frame -> True, AspectRatio -> 3/4.7, PlotRange -> All,
PlotLabel -> "A Bunch Of Burns Constants"]
giving
The terms for the MRB constant are in the green strip in the interval (-1/2,1/2). It is special, as the graphics shows clearly.
But is it sure, that the partial sums of terms on straight lines through -1 + 0 I or on straight lines through 1 + 0 I do all go to zero? Those (if any) which do not are satellites of the MRB constant.
