Time

DateString[]
NSum[(-1)^n*(n^(1/n)-1),{n,∞},WorkingPrecision->1000,Method->"AlternatingSigns"]//Timing
DateString[]

DateString[]
NSum[(-1)^n*(n^(1/n)-1),{n,∞},WorkingPrecision->314169,Method->"AlternatingSigns"]//Timing
DateString[]

DateString[]
NSum[(-1)^n*(n^(1/n)-1),{n,∞},WorkingPrecision->314169,Method->"AlternatingSigns"]//Timing
DateString[]

DateString[]
NSum[(-1)^n*(n^(1/n)-1),{n,∞},WorkingPrecision->314169,Method->"AlternatingSigns"]//Timing
DateString[]