Let m be the MRB constant, the upper limit point of the alternating series, Sum[(-1)^n n^(1/n) , {n, 1, Infinity}]:

m = NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, 
  WorkingPrecision -> 100, Method -> "AlternatingSigns"]

Consider the v'th partial sums of the MRB constant, Sum[(-1)^n (n^(1/n) - 1), {n, 1, v}].

For many natural numbers, u,the ratio

Sum[(-1)^n (n^(1/n) - 1), {n, 1, w^(x-1)}]/

Sum[(-1)^n (n^(1/n) - 1), {n, 1, w^x}]

appears to go to (x+1)/(w x) as x goes to Infinity.

Some tricky algebra in an induction proof, not done here, should confirm that it is so.

Table[N[Table[
    Ratios[Table[
        m - NSum[(-1)^n (n^(1/n) - 1), {n, 1, w^x}, 
          WorkingPrecision -> 100, Method -> "AlternatingSigns"], {x, 
         1, 50}]][[u]] - (u + 1)/(w u), {u, 1, 49}], 
   20][[40 ;; 49]], {w, 2, 5}]

gives the last 10 calculated partial sums for each w as

{{-3.0378747951200660462*10^-12, -1.5583419944145320174*10^-12, 
-7.9887326324966306249*10^-13, -4.0928774765159424587*10^-13, 
-2.0956942306321773126*10^-13, -1.0724748167264547390*10^-13, 
-5.4855123643314287025*10^-14, -2.8043252089204499055*10^-14, 
-1.4329470611601605631*10^-14, -7.3186573078356545887*10^-15}, 

  {-3.9735949593843386682*10^-19, -1.3580103284839051318*10^-19, 
-4.6382950898596934816*10^-20, -1.5832958665126716607*10^-20, 
-5.4016430216047719279*10^-21, -1.8418772359482509989*10^-21, 
-6.2773544848152706058*10^-22, -2.1383722552379956182*10^-22, 
-7.2809752368861249529*10^-23, -2.4780138536228499243*10^-23}, 

  {-4.2929163962106453177*10^-24, -1.1001134461874828148*10^-24, 
-2.8174935399220658387*10^-25, -7.2117564615326913531*10^-26, 
-1.8449442547340231345*10^-26, -4.7173722841405625970*10^-27, 
-1.2055957015698484194*10^-27, -3.0796201705616013957*10^-28, 
-7.8631261687787535433*10^-29, -2.0068001135113299619*10^-29}, 

  {-5.6792462548848094607*10^-28, -1.1641707486810064005*10^-28, 
-2.3849837529455541567*10^-29, -4.8832506380656677373*10^-30, 
-9.9930649659648556140*10^-31, -2.0439252497682916112*10^-31, 
-4.1784741112384828890*10^-32, -8.5381937554540824378*10^-33, 
-1.7438875402631071817*10^-33, -3.5602720620503867710*10^-34}}.

Which all definitely appear to go to 0.