Here is a strange way to increase the speed of NSum. A dummy second summation (below, {n, 1, 1}) nearly halves the timing!! Can anyone figure out why it works?
Can anyone at Wolfram Research figure out why?
g[x_] := (x^(1/x) - 1);
m = NSum[(-1)^x (g[x]), {x, 1, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 10000];
N[Timing[m -
NSum[(-1)^x (g[x]), {x, 1, Infinity}, Method -> "AlternatingSigns",
WorkingPrecision -> 10000, NSumTerms -> 100]], 50]
(*{88.75, 0.*10^-9998}*)
(* In[197]:=*) N[
Timing[m -
NSum[(-1)^x (g[x]), {x, 1, Infinity}, {n, 1},
Method -> "AlternatingSigns", WorkingPrecision -> 10000,
NSumTerms -> 100]], 50]
(* Out[197]= {47.7969, 0.*10^-9998}*)