Wolfram Alpha gave me several machine sized approximations of Re[D[(-1)^n*n^(1/n), n]], at a sequence of points, using named constants and 5 or less coefficients of three of less digits.:
Initialization
m = NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity},
WorkingPrecision ->
17](*A037077 Decimal expansion of upper limit \
of-1^(1/1)+2^(1/2)-3^(1/3)+.... (the MRB constant.*)
0.18785964246207
l = N[8/(3 Sqrt[3]),
15] (*LA118273 Decimal expansion of (4/3)^(3/2).Lieb's square ice constant*)
1.53960071783900
p = N[Root[#^6 - #^5 - #^4 - #^3 - #^2 - # - 1 &, 2],
15] (*A118427 Decimal expansion of hexanacci constant.*)
1.98358284342433
h9 = 0.7859336743503714545 (* oeis.org/A117239 Decimal expansion of solution \
to problem #9 in the Trefethen challenge.*)
0.785933674350371455
t = .273944195739271617171 (*A091694 Trott's second constant:decimal \
expansion is same as terms in its non-simple continued fraction read serially*)
0.27394419573927161717
g = N[ Integrate[Sin[x]/x, {x, 0, \[Pi]}],
20] (*A036792 Wilbraham-Gibbs Constant*)
1.8519370519824661704
f = x /. FindRoot[x^(1/x) - (x + 1)^(1/(x + 1)) == 0, {x, 2},
WorkingPrecision -> 20](* A085846
Decimal expansion of root of x=(1+1/x)^x. Foias' "2nd" Constant*)
2.2931662874118610315
c = N[1 - 2 Sin[Pi/18],
15](* A178959 Decimal expansion of the site percolation threshold for the \
(3,6,3,6) Kagome Archimedean lattice. Bond Percolation on Honeycomb Lattice*)
0.652703644666139
w = N[Gamma[1/3]^3/(4*Pi),
17] (*A064582 Real half-period for the Weierstrass elliptic function with \
invariants g2=0,g3=1. Weierstrass constant*)
1.5299540370571929
a = 2.50290787509589282228 (*A006891 Decimal expansion of Feigenbaum \
reduction parameter.*)
2.5029078750958928223
ff = N[(1/2^(1/3))*(-1)^(1/24)*GoldenRatio^(1/12)*
EllipticThetaPrime[1, 0, -(I/GoldenRatio)]^(1/3), 14]
1.22674201072035 + 0.*10^-15 I
no = 0.06535142592303732137 (*A143304
Decimal expansion of Norton's constant.*)
0.06535142592303732137
m2 = N[-Log[16], 16] (*A016639
Decimal expansion of log(16). Madelung constant b4(2).*)
-2.772588722239781
h8 = 0.4240113870336883638 (*A117238
Decimal expansion of solution to problem #8 in the Trefethen challenge.*)
0.4240113870336883638
s = N[2 + 2*Cos[2*Pi/7], 17] (*A116425
Decimal expansion of 2+2*cos(2*Pi/7). Silver constant.*)
3.2469796037174671
sm = N[128/(45*Pi), 17](*A093070
Decimal expansion of 128/(45*Pi). Mean line-in-disk length*)
0.90541478736722680
pe = N[Sqrt[2] InverseErf[1/2], 17] (*A092678
Decimal expansion of the probable error.*)
0.67448975019608174
ct = 0.678234491917391978035 (*A175639
Decimal expansion of product_{p=prime} (1-3/p^3+2/p^4+1/p^5-1/p^6). \
Taniguchi's Constant*)
0.67823449191739197804
an = 0.107653919226484576615323 (* A072558
Decimal expansion of One-ninth constant.*)
0.10765391922648457661532
cn = 0.6434105462883380261 (*A118227
Decimal expansion of Cahen's constant.*)
0.643410546288338026
tv = N[3/2 Sqrt[3], 15](*A020832
Decimal expansion of 1/sqrt(75). Twenty-Vertex Entropy Constant*)
2.59807621135332
jo = N[BesselJZero[0, 1], 17](*A115368
Decimal expansion of first zero of the Bessel function J_0(z).*)
2.4048255576957728
vc = N[3977/216000 - Pi^2/2160, 17] (*A093524
Decimal expansion of 3977/216000-Pi^2/2160. Mean cube-in-tetrahedron picking \
volume.*)
0.013842775740236408
Calculation
{N[Re[D[(-1)^n*n^(1/n), n] - (67 - 360*m)/(10*(9*m - 2)) /. n -> 1/E], 15],
N[Re[D[(-1)^n*n^(1/n), n] + Exp[1/E] \[Pi] Sin[E Pi] /. n -> E], 20]}
N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating Re[I (-1)^E E^(1/E) \[Pi]]+E^(1/E) \[Pi] Sin[E \[Pi]].
{-9.2*10^-11, 0.*10^-69}
{N[Re[D[(-1)^n*n^(1/n), n] - 9/(520*l + 279) /. n -> m], 15],
N[Re[D[(-1)^n*n^(1/n)*n] + (99 - 94*h8)/(41*h8 - 2) /. n -> 1/m], 15]}
{7.1*10^-13, 2.93*10^-10}
{N[Re[D[(-1)^n*n^(1/n), n] + (2*(31*p + 110))/(81*p - 7) /. n -> 1/l], 15],
N[Re[D[(-1)^n*n^(1/n), n] - (86*s + 401)/(3*(103*s - 280)) /. n -> l], 15]}
{-8.4*10^-12, -1.7*10^-13}
{N[Re[D[(-1)^n*n^(1/n), n]] - (2*(50*h9 + 3))/(3*(91*h9 + 11)) /. n -> p, 15],
N[Re[D[(-1)^n*n^(1/n), n]] - (2*(13*ct + 2))/(22*ct - 41) /. n -> 1/p, 15]}
{1.5*10^-12, 9.0*10^-13}
{N[Re[D[(-1)^n*n^(1/n), n] + 169/(236*t) - 1/59 /. n -> h9], 15],
{N[Re[D[(-1)^n*n^(1/n), n] - (2 + 4*sm - 419/134) /. n -> 1/h9], 15],
N[Re[D[(-1)^n*n^(1/n), n] - (2 + (30 - 343*an)/(2*(353*an - 45))) /.
n -> 1/h9], 15]}}
{-3.5044*10^-13, {-2.2479*10^-12, 6.84515*10^-12}}
{N[Re[D[(-1)^n*n^(1/n), n] - (20*(140*g + 3))/(1041*g - 608) /. n -> 1/t],
15],
N[Re[D[(-1)^n*n^(1/n), n] - (170*cn + 7)/(2*(2*cn + 373)) /. n -> t], 15]}
{1.09500*10^-13, -2.2315*10^-15}
{N[Re[D[(-1)^n*n^(1/n), n] + (10*(7*f + 29))/(208*f - 691) /. n -> g], 15],
N[Re[D[(-1)^n*n^(1/n), n] + (969 tv - 547)/(100 (7 tv - 2)) /. n -> 1/g],
16]}
{-6.78824*10^-13, -3.*10^-14}
{N[Re[D[(-1)^n*n^(1/n), n] + (4*(135*c + 17))/(193*c - 8) /. n -> f], 16],
N[Re[D[(-1)^n*n^(1/n), n] + (-200 jo - 27)/(15 (5 jo - 208)) /. n -> 1/ f],
18]}
{-2.6*10^-13, 3.2*10^-16}
{N[Re[D[(-1)^n*n^(1/n), n] + 228/(773*m - 200) /. n -> 1/c], 16],
N[Re[D[(-1)^n*n^(1/n) n] + (-5 c - 3)/(4 (95 c - 72)) /. n -> c] , 16]}
{-8.*10^-13, -1.61*10^-12}
{N[Re[D[(-1)^n*n^(1/n), n] - (6 - (376*w)/247) /. n -> 1/m], 17],
N[Re[D[(-1)^n*n^(1/n), n] + 1 + (5 (89 pe - 80))/(2 (23 pe + 34)) /.
n -> m], 17]}
{-4.*10^-12, -1.82*10^-12}
{N[Re[D[(-1)^n*n^(1/n), n] - (10*(667*a + 37))/(3000*a - 3407) /. n -> w],
19], N[Re[D[(-1)^n*n^(1/n), n] + 100 (62 vc - (87 E)/283) /. n -> 1/w],
19]}
{0.*10^-17, 7.684*10^-12}
{N[Re[D[(-1)^n*n^(1/n), n] - 829/11100 /. n -> 1/a], 13],
N[Re[D[(-1)^n*n^(1/n), n] - (2*(7*ff + 45))/(103*ff - 150) /. n -> a], 13]}
{-5.37550888*10^-11, -1.28*10^-10}
{N[Re[D[(-1)^n*n^(1/n), n] + (7*(no - 3))/(467*no - 20) /. n -> ff], 13],
N[Re[D[(-1)^n*n^(1/n), n] + (43*m2 - 72)/(5*m2 - 62) /. n -> 1/ff], 13]}
{-2.35*10^-10, 2.7*10^-11}
{N[Re[D[(-1)^n*n^(1/n), n] /. n -> no], 20],
Limit[Re[D[(-1)^n*n^(1/n), n]], n -> 0]}
{6.3472949791887308*10^-16, 0}
{N[Re[D[(-1)^n*n^(1/n), n] + 95 + (10*(sm + 2))/(24*sm + 49) /. n -> 1/m2],
15],
N[Re[D[(-1)^n*n^(1/n), n] + (95*pe)/169 + E/7 /. n -> m2], 15]}
{5.*10^-11, 5.1*10^-13}
