Which test for Series Convergence/Divergence?

Question

I'm after some advice please... Given the following series :

$$ \sum_{n=2}^{\infty} \frac{(ln(n+1)+n)^a}{ln^2(n)}, a>1 $$

What would be the appropriate test to perform to find whether is it Convergent or not? I'm thinking an Integral Test but would a Comparison Test be better?

Any pointers appreciated, thankyou.

Answer 1

Note that we don't have the necessary condition for convergence $a_n\to 0$, indeed

$$\frac{(\ln(n+1)+n)^a}{\ln^2(n)}=n^a\frac{(\frac{\ln(n+1)}{n}+1)^a}{\ln^2(n)}\to+\infty$$

Answer 2

$\ln n\leq n^{1/4}$ for large $n$, then $(\ln n)^{2}\leq n^{1/2}$ and hence $\dfrac{(\ln(n+1)+n)^{a}}{(\ln n)^{2}}\geq\dfrac{n}{(\ln n)^{2}}\geq n^{1/2}$.