Looking only at convergent series of the form Sum[(-1)^k (k^(1/k) - f(k)), {k, 1, Infinity}], I found the following.
Let m be the MRB constant,
m=NSum[(-1)^k (k^(1/k) - 1), {k, 1, Infinity}, WorkingPrecision -> 30, Method -> "AlternatingSigns"];
. Then
NSum[(-1)^k (k^(1/k) - Tanh[k]), {k, 1, Infinity}, WorkingPrecision -> 30, Method -> "AlternatingSigns"] - (10 (4 - 27 m))/(407 m + 490)
=2.397493242614054*10^-14 .
For an approximation, I think that is pretty notable!
If you think so too, pass it on, please.
