# Wolfram says sum diverges, but Mathematica gives a numerical value for infinite sum [closed]

Take this sum for example: $$\sum_{n=2}^\infty\frac1{\log(n!)}$$ Wolfram says that this does not converge by the comparison test. However, when I use Mathematica's NSum function, it returns a numerical value for the summation. Who should I trust?

NSum[1/Log[n!], {n, 2, \[Infinity]}]=6.12902

• Stirling: $\log n!\sim n\log n$, and $\frac{1}{n\log n}$ is not summable. – AccidentalFourierTransform Nov 3 '18 at 3:35
• @AccidentalFourierTransform by not summable, do you mean divergent? – John Glenn Nov 3 '18 at 3:36
• You should have mentioned that Mathematica throws error messages when executing NSum[1/Log[n!], {n, 2, \[Infinity]}]... – Henrik Schumacher Nov 3 '18 at 7:05
• From the doc page for NSum under the section Possible Issues: "NSum may not detect divergence for some infinite sums" and "Convergence verification is based on a ratio test that is inconclusive when equal to 1", both of which apply here. – Michael E2 Nov 3 '18 at 13:55
• I made test PHP script which evaluates numerically this series. My results are n=23,166,000,000; sum=6.2010890765879; term_n=1.8878160302708E-12. So clearly a bigger value than mathematica's 6.12902, but it's not surprising, because series converges very slowly and mathematica doesn't have a patience :-) – Agnius Vasiliauskas Nov 6 '18 at 14:10

When a command has issues, it is a good idea to check the Possible Issues section of the doc page for the command, in this case NSum. Two of the issues apply to the OP's case:
NSum may not detect divergence for some infinite sums