For which $a , b , c$ $\sum \frac{a^n}{(\log n)^c n^b}$ would be divergent?

Question

$$\sum \frac{a^n}{(\log n)^c n^b}$$

Can anyone please explain me what will be the range of $a , b , c$ for the divergence of that series?

My TrY : I think when $|a| > 1$ then for any $b ,c $ the series will be divergent and $|a| < 1$ then for any $b ,c $ the series will be convergent. But I have NO idea about the case when $|a| = 1$.

Please correct me If I went wrong anywhere.

Thank You in Advace.

Answer 1

HINT

For the case $a=1$ refer to Cauchy condensation test.

For the case $a=-1$ refer to alternating series test.