# Calculate 40 digits of the MRB constant

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?

NSum[(-1)^k (k^(1/k) - 1), {k, 1, ∞},

• You might want to try the setting Method -> "AlternatingSigns", so you don't need a large setting of NSumTerms. – J. M.'s technical difficulties Oct 25 '17 at 8:50