f[x_]=(-1)^x (x^(1/x)-1);

CMRB=NSum[f[x],{x,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->100]

Here are the (sums as k->∞) of CMRB=(∑_(x=1)^∞)(((((-1))^x) log(x) (∑_(n=1)^(k-1))((k! (x^(k-n-1)) (log^(n-1))(x))/(k n!)))/((k-1)! (x^(k-1)))):

(l=Table[Sum [(1/((k-1)! (x^(k-1))))(-1)^x Log[x] Sum[k!/(n)!x^(k-1-n)Log[x]^(n-1)/k,{n,1,k-1}],{x,1,Infinity}],{k,1,5}])//TableForm

l-CMRB

The differences those partial sums of CMRB telescope nicely, as follows.

result1=(Differences[l])//Expand//TableForm

cf. Table[(∑_∞^(x=1))(((E^(I π x)) (log^n)(x))/(n! (x^n))),{n,1,5}]

(ll=Table[(Sum[E^(I π x) Log[x]^n/(n! x^n), {x, 1, Infinity}]), {n, 
    1, 5}])//TableForm

result2=(Differences[l])//Expand//TableForm

result2-result1

ll-l