MRB constant is the upper limit point of the following sequence
$$s_n=\sum_{k=1}^{n} (-1)^k k^{\frac{1}{k}}$$
$MRB=\color{blue}{0.1878596}...$
I tried to calculate first few digits:
Sum[(-1)^k k^(1/k), {k, 1, 3 000 000}] // N // AbsoluteTiming
$\lbrace{83.152 , \color{blue}{0.1878}} \color{red}{62} \rbrace $
I also tried NSum
but ... there is a different result
NSum[(-1)^k k^(1/k), {k, 1, Infinity}]
$\color{red}{-0.31214}$
Block[{$MaxExtraPrecision = 1000},
NSum[(-1)^k k^(1/k), {k, 1, Infinity}, WorkingPrecision -> 40]]
How can I calculate 40 digits of the MRB constant?