You can take a look at the Bertrand Series1. [Sigma n=2] 1 / (n * sqrt(ln(n))
In
1.
If a>1 , the Bertrand series converges regardless of the value of b or;
If a<1 , the Bertrand series diverges regardless of the value of b or;
If a=1 , the Bertrand series converges if and only if b>1 .