Wolfram says sum diverges, but Mathematica gives a numerical value for infinite sum

As the message shows, NSum works on limited recursions. Mathematica does not consider the "mathematical converge" when it works on NSum. Therefore Mathematica finds that this sum converges too slowly, and threw out the answer after MaxRecursion.

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

Convergence verification is based on a ratio test that is inconclusive when equal to 1

[Edit notice: I made a mistake in the integral test which I've deleted.]