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.