In[]:=
m=NSum[
n
(-1)
(
1/n
n
-1),{n,1,Infinity},WorkingPrecision->70,Method->"AlternatingSigns"]
Out[]=
0.18785964246206712024851793405427323005590309490013878617200468547342
In[]:=
m-NSum[(-1)^k(k^(1/k)-1),{k,1,Infinity},WorkingPrecision->100,NSumTerms->10000]
Out[]=
0.×
-44
10