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Hint: sqrt(n)/n = 1/sqrt(n)
You don't get 0/0. Simplify it.
Let a = ln(n)/n2. The series ln(n)/n2 converges if limit when n goes to infinity of ln(1/a)/ln(n) > 1
You could try and use a maclaurin sequence for ln(1+u) where u = n-1. Then after you expand a few terms of your series in terms of u, it might start to become clear where terms are going to start canceling when divided by n2.
log n is o(nε) for every ε>0 so sum log n/ns converges for s>1