Find nature of series

Question

We have $$\sum_{n=1}^{\infty}\frac{n^{\ln(n+1)}}{n^{\ln(an+1)}},$$ and problem asks for nature of that series, discussed after values of parameter $a$. I tried with D'Alembert method but seems to become so complicated and get no result.

Answer 1

$$\log(n+1)-\log(an+1)=\log\left(1+\dfrac1n\right)-\log\left(1+\dfrac1{an}\right)-\log a \\=\left(1-\frac1a\right)\frac1n+o\left(\dfrac1n\right)-\log a.$$

Then the general term is asymptotic to $n^{-\log a}$ (because $\sqrt[n]n$ tends to $1$), for which the convergence condition is known.