If $\sum a_n$ diverges then $\sum \frac{a_n}{\ln n}$ diverges If $\sum a_n$ diverges does $\sum \frac{a_n}{\ln n}$ necessarily diverge for $a_n>0$?
I tried $a_n=\frac{\ln n}{n^2}$ to try bag an easy counterexample but it turns out $\sum a_n$ converges(even things like $\frac{\ln n}{n^{1.0001}} $ converge so Id suspect this entire class converges). I also tried things like $a_n=\ln \ln \ln \ln n$ but even for these, both sums diverge. Don't tell me its true?

No this is not true.
For example $\sum\frac{1}{n\cdot \ln n}$ diverges, but $\sum\frac{\frac{1}{n\cdot \ln n}}{\ln n}=\sum\frac{1}{n\cdot (\ln n)^2}$ converges by the integral test.
