I'm studying for a calcus exam and I'm trying to solve some of the proposed exercises.
I need to study the convergence for $|x+1|^5=R$ $$\sum_{n=0}^{+\infty} \frac{\log(n+12)}{(n+12)\cdot 3^n}\cdot (x+1)^{5n}.$$
The limit of $a_n/a_{n+1}$ (using D'Alembert formula) gives me $R$ $$\lim_{n\to\infty} \frac{\log(n+12)}{(n+12)\cdot 3^n})\cdot(x+1)^{5n}\left(\frac{\log(n+1+12)}{(n+1+12)\cdot 3^{n+1}}\cdot(x+1)^{5(n+1)}\right)^{—1} = 3/(1+x)^5$$ But now I'm not sure what I need to do in order to study the convergence and I'd love a few pointers.