Determine convergence or divergence of the following series

Question

$$\sum _{ n=10 }^{ \infty }{ (-1)^ n } \frac { n^ 2 }{ \ln(n) }$$

My attempt

Since this is an alternating series,

$$\lim _{ n\rightarrow \infty }{ \frac { n^ 2 }{ \ln(n) } } $$

diverges. So, the series diverges. Can anyone please verify this?

Answer 1

I don't have enough rep to comment, so I'll write this as an answer. Yes, you're correct.

Since $\lim_n \frac{n^2}{\log n} = \infty$, it follows that $\lim_n (-1)^n\frac{n^2}{\log n} \neq 0$. Thus, the series diverges.