If I opened the right windows, the Wolfram|Alpha command had an overall 'divided by n,' while the Mathematica command didn't.
That's strange. If I make the same exact Wolfram|Alpha query from within Mathematica the answer comes out correct.
I haven't thought of this but for k=2 the problem persists, and
NSum[Log[Sqrt[k], k], {k, 2, 3}]
is defined
There is a problem in sequence B when k=1, because the logarithm of 1 in base 1 is not defined. Try
k=1
NSum[Log[Sqrt[k], k], {k, 1, 3}]
This may be at the heart of the problem, but I don't know the exact mechanism.
