Absolute or conditional convergence: $\sum_{n=1}^\infty (-1)^n \frac {2n^2+3n+4}{2n^4+3}$?

Question

Determine whether the series:

$$\sum_{n=1}^\infty (-1)^n \frac {2n^2+3n+4} {2n^4 + 3}$$

converges absolutely, conditionally or diverges.

I know the series converges conditionally using alternating series test.

My question is how do I determine absolute convergence here.

I tried limit comparison test with $\frac{1}{n^2}$ which results in conditional convergence only, is that the way to go or not?

Answer 1

Since$$\left\lvert(-1)^n\frac{2n^2+3n+4}{2n^4+3}\right\rvert=\frac{2n^2+3n+4}{2n^4+3}$$and since$$\lim_{n\to\infty}\frac{\frac{2n^2+3n+4}{2n^4+3}}{\frac1{n^2}}=1,$$one deduces from the comparison test that the series converges absolutely.