The convergent sum for MRB constant, as defined at MathWorld is approximated by

NSum[(-1)^n *((n)^(1/n) - 1), {n, 1, Infinity}, WorkingPrecision -> 30, Method -> "AlternatingSigns"]

=0.18785964246206712024857897184.

The convergent sum for Generalized MRB constant is approximated by

NSum[(I)^n *((-n)^(1/n) - 1), {n, 1, Infinity}, WorkingPrecision -> 30, Method -> "AlternatingSigns"]

=1.5799359798382254715673492184 - 2.7085160796669495851478742534 I Its real, imaginary and module parts are all close to Pi/2, -E and Pi, to 91/10,000, 97/10,000 and 59/10,000 respectively.

  NSum[(-1)^n *((n)^(1/n) - 1), {n, 1, Infinity}, 
 WorkingPrecision -> 30, Method -> "AlternatingSigns"]

 (* 0.18785964246206712024857897184*)

 Sum[(I)^n *((-n)^(1/n) - 1), {n, 1, Infinity}]

$$\sum _{n=1}^{\infty } i^n \left((-n)^{1/n}-1\right)$$

NSum[(I)^n *((-n)^(1/n) - 1), {n, 1, Infinity}, 
 WorkingPrecision -> 30, Method -> "AlternatingSigns"]

(* 1.5799359798382254715673492184 - 
 2.7085160796669495851478742534 I*)

(Pi/2 - E I) - 
 NSum[(I)^n *((-n)^(1/n) - 1), {n, 1, Infinity}, 
  WorkingPrecision -> 30, Method -> "AlternatingSigns"]

(* -0.0091396530433288523360275268 - 
 0.0097657487920956502124132179 I*)

 Pi - 
 Abs[NSum[(I)^n *((-n)^(1/n) - 1), {n, 1, Infinity}, 
   WorkingPrecision -> 30, Method -> "AlternatingSigns"]]

(* 0.0059496486860469938011013043*)