Let $\sum a_n,\sum b_n\in\mathbb{R}_+$ be two series and let $F(x),G(x)$ be their corresponding generating functions, that is, $$ F(x)=\sum_{n=0}^\infty a_nx^n\\ G(x)=\sum_{n=0}^\infty b_nx^n$$ In addition assume that $F(1),G(1)<\infty$. Then, if one would want to estimate the Cauchy product $c_n=\sum_{k=0}^n a_k b_{n-k}$ one could use its generating function $F(x)\cdot G(x)$ . Assuming that the radius of convergence is $r_0>1$ we get the following estimation for any $\epsilon>0$: $$|c_n|\leq\Big(\frac{1}{r_0}+\epsilon\Big)^n $$
I would like to ask the following question: can one estimate the Cauchy product of two series when one of the series diverges but the other one converges (and perhaps get an estimate which is not geometric)? In particular I'm interested in the case when $a_n=q^n, b_n=\frac{1}{n+1}$ for $q\in (0,1)$ and $$ F(x)=\frac{1}{1-qx} \\ G(x)=-\frac{log(1-x)}{x} $$ and then $$ \sum_{n=0}^\infty c_n x^n=-\frac{log(1-x)}{x-qx^2} $$ How fast the coefficients $c_n$ tend to zero if at all?